INTERNATIONAL GONGRESS bounded by (2.5) for all ℓâ„“â„“\ellâ„“; and infinitely many totally real degree nSnnSnnS_(n)n S_{n}nSn-fields KKKKK with ClKClKCl_(K)\mathrm{Cl}_{K}ClK containing an element of exact order ℓâ„“â„“\ellâ„“ and |ClK[ℓ]|ClK[â„“]|Cl_(K)[â„“]|\left|\mathrm{Cl}_{K}[\ell]\right||ClK[â„“]| bounded by (2.5).) What happens when GGGGG does not have a unique minimal nontrivial normal subgroup? Here is an open question: in general, when NNNNN is a nontrivial normal subgroup of GGGGG (not necessarily unique or minimal), what is the true order of growth of (4.12) as X→∞X→∞X rarr ooX \rightarrow \inftyX→∞ ? Questions about this "intersection multiplicity" are gathered in [62].
Third, increased attention has turned to bounding ℓâ„“â„“\ellâ„“-torsion in class groups for all fields in special families specified by the Galois group: that is, proving property CF,ℓ(Δ)CF,â„“(Δ)C_(F,â„“)(Delta)\mathbf{C}_{\mathscr{F}, \ell}(\Delta)CF,â„“(Δ) for some Δ<1/2Δ<1/2Delta < 1//2\Delta<1 / 2Δ<1/2. First, Klüners and Wang have proved CF,p(0)CF,p(0)C_(F,p)(0)\mathbf{C}_{\mathscr{F}, p}(0)CF,p(0) for the family Fpr(G;X)Fpr(G;X)F_(p^(r))(G;X)\mathscr{F}_{p^{r}}(G ; X)Fpr(G;X) for any ppppp-group GGGGG; this generalizes the application of genus theory to prove C2,2(0)C2,2(0)C_(2,2)(0)\mathbf{C}_{2,2}(0)C2,2(0) [54].
Second, let G=(Z/pZ)rG=(Z/pZ)rG=(Z//pZ)^(r)G=(\mathbb{Z} / p \mathbb{Z})^{r}G=(Z/pZ)r be an elementary abelian group of rank r≥2r≥2r >= 2r \geq 2r≥2, with ppppp prime. Wang has shown that for every ℓâ„“â„“\ellâ„“, within the family of Galois GGGGG-fields K/QK/QK//QK / \mathbb{Q}K/Q, property CF,ℓ(1/2−δ(ℓ,p))CF,â„“(1/2−δ(â„“,p))C_(F,â„“)(1//2-delta(â„“,p))\mathbf{C}_{\mathscr{F}, \ell}(1 / 2-\delta(\ell, p))CF,â„“(1/2−δ(â„“,p)) holds for some δ(ℓ,p)>0δ(â„“,p)>0delta(â„“,p) > 0\delta(\ell, p)>0δ(â„“,p)>0 [91]. Since the savings δ(ℓ,p)δ(â„“,p)delta(â„“,p)\delta(\ell, p)δ(â„“,p) is independent of the rank, for rrrrr sufficiently large this is better than CF,ℓ(ΔGRH)CF,ℓΔGRHC_(F,â„“)(Delta_(GRH))\mathbf{C}_{\mathscr{F}, \ell}\left(\Delta_{\mathrm{GRH}}\right)CF,â„“(ΔGRH). The method of proof plays off the interaction of three facts arising from the precise structure of GGGGG : first, |ClK[ℓ]|ClK[â„“]|Cl_(K)[â„“]|\left|\mathrm{Cl}_{K}[\ell]\right||ClK[â„“]| factors as a product of |ClF[ℓ]|ClF[â„“]|Cl_(F)[â„“]|\left|\mathrm{Cl}_{F}[\ell]\right||ClF[â„“]| where FFFFF varies over the ≈pr−1≈pr−1~~p^(r-1)\approx p^{r-1}≈pr−1 many degree ppppp subfields of KKKKK, so it suffices to bound one of these factors nontrivially. Second, any rational prime splits completely in ≈pr−2≈pr−2~~p^(r-2)\approx p^{r-2}≈pr−2 of these subfields, so at least one subfield has a positive proportion of primes splitting completely in it. Third, the sizes of the discriminants of the subfields can be played against each other, so that known prime-counting results (which may a priori seem to count primes that are "too large") suffice for the application of the Ellenberg-Venkatesh criterion. This is an interesting counterpoint to the methods described earlier. In another direction, Wang has developed the notion of a forcing extension; certain nilpotent groups can be built from elementary ppppp-groups via forcing extensions. If G′G′G^(')G^{\prime}G′ is constructed from GGGGG by a forcing extension, then CF′,ℓ(Δ′)CF′,ℓΔ′C_(F^('),â„“)(Delta^('))\mathbf{C}_{\mathscr{F}^{\prime}, \ell}\left(\Delta^{\prime}\right)CF′,â„“(Δ′) can be deduced from CF,ℓ(Δ)CF,â„“(Δ)C_(F,â„“)(Delta)\mathbf{C}_{\mathscr{F}, \ell}(\Delta)CF,â„“(Δ), for some Δ,Δ′<1/2Δ,Δ′<1/2Delta,Delta^(') < 1//2\Delta, \Delta^{\prime}<1 / 2Δ,Δ′<1/2, where FFF\mathscr{F}F is the family of GGGGG-extensions and F′F′F^(')\mathscr{F}^{\prime}F′ is the family of G′G′G^(')G^{\prime}G′-extensions [89].
All of the results mentioned in this section (except where genus theory suffices) directly apply or build on the Ellenberg-Venkatesh criterion. Can this criterion be strengthened? Ellenberg has suggested some possible improvements in [32]. In particular, let η(K):=inf{HK(α):K=Q(α)}η(K):=infHK(α):K=Q(α)eta(K):=i n f{H_(K)(alpha):K=Q(alpha)}\eta(K):=\inf \left\{H_{K}(\alpha): K=\mathbb{Q}(\alpha)\right\}η(K):=inf{HK(α):K=Q(α)} denote the minimum (relative) multiplicative Weil height of a generating element of KKKKK. Roughly speaking, Ellenberg notes the criterion (2.4) can actually allow prime ideals with norms as large as η(K)1/ℓη(K)1/â„“eta(K)^(1//â„“)\eta(K)^{1 / \ell}η(K)1/â„“. The restriction to norms <DK12ℓ(n−1)<DK12â„“(n−1)< D_(K)^((1)/(2â„“(n-1)))<D_{K}^{\frac{1}{2 \ell(n-1)}}<DK12â„“(n−1) in (2.4) was made since the lower bound η(K)≥DK12(n−1)η(K)≥DK12(n−1)eta(K) >= D_(K)^((1)/(2(n-1)))\eta(K) \geq D_{K}^{\frac{1}{2(n-1)}}η(K)≥DK12(n−1) holds for all fields [80]. Widmer, also with Frei, has shown that η(K)η(K)eta(K)\eta(K)η(K) can be enlarged for almost all fields in certain families, leading to improved upper bounds for ℓâ„“â„“\ellâ„“-torsion in those fields [41,95]. That is, they improve the very notion of the "GRH-bound" (2.5), and show that the parameter we have called ΔGRH ΔGRH Delta_("GRH ")\Delta_{\text {GRH }}ΔGRH can actually be taken smaller for some fields. Their work raises interesting open questions: what upper and lower bounds hold for η(K)η(K)eta(K)\eta(K)η(K), for all (or almost all) fields in a family? Ruppert [74] has conjectured uniform upper bounds η(K)≤DK1/2η(K)≤DK1/2eta(K) <= D_(K)^(1//2)\eta(K) \leq D_{K}^{1 / 2}η(K)≤DK1/2 (now proved for almost all fields in some families by [72]). If this is true, the Ellenberg-Venkatesh criterion would hit a barrier, for most fields, with a result like |ClK[ℓ]|≪DK1/2−1/2ℓ+εClK[â„“]≪DK1/2−1/2â„“+ε|Cl_(K)[â„“]|≪D_(K)^(1//2-1//2â„“+epsi)\left|\mathrm{Cl}_{K}[\ell]\right| \ll D_{K}^{1 / 2-1 / 2 \ell+\varepsilon}|ClK[â„“]|≪DK1/2−1/2â„“+ε for any degree nnnnn, still far from the ℓâ„“â„“\ellâ„“-torsion Conjecture. It would be very interesting to find a new, different criterion.
5. WHY DO WE EXPECT THE ℓâ„“â„“\ellâ„“-TORSION CONJECTURE TO BE TRUE?
Recall that the ℓâ„“â„“\ellâ„“-torsion Conjecture 2.1 is still known only in the case stemming from Gauss's work, namely for n=2,ℓ=2n=2,â„“=2n=2,â„“=2n=2, \ell=2n=2,â„“=2. It is a good idea to affirm why we believe the ℓâ„“â„“\ellâ„“-torsion Conjecture should be true. We will consider this from three perspectives.
5.1. From the perspective of the Cohen-Lenstra-Martinet heuristics
So far, when we have mentioned a result for almost all fields in a family, we have not focused on the size of a potential exceptional set, other than showing it is smaller than the size of the full family. But to understand the ℓâ„“â„“\ellâ„“-torsion Conjecture, we must quantify a potential exceptional set, and show that for all sufficiently large discriminants, it is empty.
Let us abstract this, for a family F0(X)F0(X)F_(0)(X)\mathscr{F}_{0}(X)F0(X) of fields KKKKK with DKDKD_(K)D_{K}DK in a dyadic range (X/2,X](X/2,X](X//2,X](X / 2, X](X/2,X], from which more general results can easily be deduced by summing over ≪logX≪logâ¡X≪log X\ll \log X≪logâ¡X dyadic ranges. Suppose f:F0(X)→Nf:F0(X)→Nf:F_(0)(X)rarrNf: \mathscr{F}_{0}(X) \rightarrow \mathbb{N}f:F0(X)→N is a function with f(K)≤DKaf(K)≤DKaf(K) <= D_(K)^(a)f(K) \leq D_{K}^{a}f(K)≤DKa for all KKKKK. Suppose that for some Δ<aΔ<aDelta < a\Delta<aΔ<a we can improve this to f(K)≤DKΔf(K)≤DKΔf(K) <= D_(K)^(Delta)f(K) \leq D_{K}^{\Delta}f(K)≤DKΔ for all KKKKK outside of some exceptional set E0Δ(X)⊂F0(X)E0Δ(X)⊂F0(X)E_(0)^(Delta)(X)subF_(0)(X)E_{0}^{\Delta}(X) \subset \mathscr{F}_{0}(X)E0Δ(X)⊂F0(X). Then
As long as |E0Δ(X)|≪|F0(X)|X−(a−Δ)E0Δ(X)≪F0(X)X−(a−Δ)|E_(0)^(Delta)(X)|≪|F_(0)(X)|X^(-(a-Delta))\left|E_{0}^{\Delta}(X)\right| \ll\left|\mathscr{F}_{0}(X)\right| X^{-(a-\Delta)}|E0Δ(X)|≪|F0(X)|X−(a−Δ), this shows that f(K)≪XΔf(K)≪XΔf(K)≪X^(Delta)f(K) \ll X^{\Delta}f(K)≪XΔ on average. On the other hand, suppose we know ∑K∈F0(X)f(K)≤Xb∑K∈F0(X) f(K)≤Xbsum_(K inF_(0)(X))f(K) <= X^(b)\sum_{K \in \mathscr{F}_{0}(X)} f(K) \leq X^{b}∑K∈F0(X)f(K)≤Xb. Then a potential set of exceptions E0Δ(X)={K∈F0(X):f(K)>DKΔ}E0Δ(X)=K∈F0(X):f(K)>DKΔE_(0)^(Delta)(X)={K inF_(0)(X):f(K) > D_(K)^(Delta)}E_{0}^{\Delta}(X)=\left\{K \in \mathscr{F}_{0}(X): f(K)>D_{K}^{\Delta}\right\}E0Δ(X)={K∈F0(X):f(K)>DKΔ} can be controlled by
Thus |E0Δ(X)|≪Xb−ΔE0Δ(X)≪Xb−Δ|E_(0)^(Delta)(X)|≪X^(b-Delta)\left|E_{0}^{\Delta}(X)\right| \ll X^{b-\Delta}|E0Δ(X)|≪Xb−Δ, and exceptional fields are density zero in F0(X)F0(X)F_(0)(X)\mathscr{F}_{0}(X)F0(X), provided Xb−Δ=o(|F0(X)|)Xb−Δ=oF0(X)X^(b-Delta)=o(|F_(0)(X)|)X^{b-\Delta}=o\left(\left|\mathscr{F}_{0}(X)\right|\right)Xb−Δ=o(|F0(X)|). That is, a nontrivial upper bound on ℓâ„“â„“\ellâ„“-torsion for "almost all" fields in a family FFF\mathscr{F}F is essentially equivalent to the same upper bound "on average."
To verify the ℓâ„“â„“\ellâ„“-torsion Conjecture, we wish to show a "pointwise" bound: for every ε>0ε>0epsi > 0\varepsilon>0ε>0, there exists DεDεD_(epsi)D_{\varepsilon}Dε such that when DK≥DεDK≥DεD_(K) >= D_(epsi)D_{K} \geq D_{\varepsilon}DK≥Dε, there are no exceptions to the bound |ClK[ℓ]|≤DKεClK[â„“]≤DKε|Cl_(K)[â„“]| <= D_(K)^(epsi)\left|\mathrm{Cl}_{K}[\ell]\right| \leq D_{K}^{\varepsilon}|ClK[â„“]|≤DKε. The key is to consider not averages but arbitrarily high kkkkk th moments. In the general setting above, suppose that we know ∑K∈F0(X)f(K)k≤Xb∑K∈F0(X) f(K)k≤Xbsum_(K inF_(0)(X))f(K)^(k) <= X^(b)\sum_{K \in \mathscr{F}_{0}(X)} f(K)^{k} \leq X^{b}∑K∈F0(X)f(K)k≤Xb, for a real number k≥1k≥1k >= 1k \geq 1k≥1. Then for any fixed Δ>0Δ>0Delta > 0\Delta>0Δ>0, adapting the argument (5.2) shows that |E0Δ(X)|≪Xb−kΔE0Δ(X)≪Xb−kΔ|E_(0)^(Delta)(X)|≪X^(b-k Delta)\left|E_{0}^{\Delta}(X)\right| \ll X^{b-k \Delta}|E0Δ(X)|≪Xb−kΔ. If the kkkkk th moment is uniformly bounded by XbXbX^(b)X^{b}Xb for a sequence of k→∞k→∞k rarr ook \rightarrow \inftyk→∞, then for each Δ>0Δ>0Delta > 0\Delta>0Δ>0, we can take kkkkk sufficiently large to conclude that the set of exceptions is empty.
This perspective has been applied by Pierce, Turnage-Butterbaugh, and Wood in [73] to prove that the ℓâ„“â„“\ellâ„“-torsion Conjecture holds for all fields in a family F(X)F(X)F(X)\mathscr{F}(X)F(X) if there is a real number α≥1α≥1alpha >= 1\alpha \geq 1α≥1 such that for a sequence of arbitrarily large kkkkk,
(5.3)∑K∈F(X)|ClK[ℓ]|k≪n,ℓ,k,α|F(X)|α, for all X≥1(5.3)∑K∈F(X) ClK[â„“]k≪n,â„“,k,α|F(X)|α, for all X≥1{:(5.3)sum_(K inF(X))|Cl_(K)[â„“]|^(k)≪_(n,â„“,k,alpha)|F(X)|^(alpha)","quad" for all "X >= 1:}\begin{equation*}
\sum_{K \in \mathscr{F}(X)}\left|\mathrm{Cl}_{K}[\ell]\right|^{k} \ll_{n, \ell, k, \alpha}|\mathscr{F}(X)|^{\alpha}, \quad \text { for all } X \geq 1 \tag{5.3}
\end{equation*}(5.3)∑K∈F(X)|ClK[ℓ]|k≪n,ℓ,k,α|F(X)|α, for all X≥1
The Cohen-Lenstra-Martinet heuristics predict that (5.3) holds, in the form of an even stronger asymptotic with α=1α=1alpha=1\alpha=1α=1, for all integers k≥1k≥1k >= 1k \geq 1k≥1, for families of Galois GGGGG-extensions, at least for all primes ℓ∤|G|ℓ∤|G|ℓ∤|G|\ell \nmid|G|ℓ∤|G|. The appropriate moment formulation can be found in [21] for degree 2 fields and in [92] for higher degrees, building on [22]. This confirms that the ℓâ„“â„“\ellâ„“-torsion Conjecture follows from another well-known set of conjectures.
The Cohen-Lenstra-Martinet heuristics are a subject of intense interest and much recent activity. Here are some spectacular successes most closely related to our topic. Dav-
enport and Heilbronn [28] have proved
in which each isomorphism class of fields is counted once. Very recently, [63] obtained analogues of (5.4) for averages over F2m(G;X)F2m(G;X)F_(2^(m))(G;X)\mathscr{F}_{2^{m}}(G ; X)F2m(G;X) for any permutation group G⊂S2mG⊂S2mG subS_(2^(m))G \subset S_{2^{m}}G⊂S2m that is a transitive permutation 2-group containing a transposition. See also the work of Smith on the distribution of 2k2k2^(k)2^{k}2k-class groups in imaginary quadratic fields [81]; Koymans and Pagano on ℓkâ„“kâ„“^(k)\ell^{k}â„“k-class groups of degree ℓâ„“â„“\ellâ„“ cyclic fields [59]; Klys on moments of ppppp-torsion in cyclic degree ppppp fields (conditional on GRH for p≥5p≥5p >= 5p \geq 5p≥5 ) [55]; Milovic and Koymans on 16-rank in quadratic fields [57,58]; Bhargava and Varma [13,14] elaborating on (5.4) and (5.5).
The perspective of moments (5.3) provides a strong motivation to prove the kkkkk th moment bounds for ℓâ„“â„“\ellâ„“-torsion. Fouvry and Klüners have proved an asymptotic for the kkkkk th moments related to 4-torsion when KKKKK is quadratic, for all integers k≥1k≥1k >= 1k \geq 1k≥1 [38]. Heath-Brown and Pierce have proved nontrivial bounds for the kkkkk th moments of ℓâ„“â„“\ellâ„“-torsion for imaginary quadratic fields, for all odd primes ℓâ„“â„“\ellâ„“ [46]. For example, they establish second moment bounds
(5.6)∑K=Q(±D)D≤X|ClK[3]|2≪X23/18,∑K=Q(−D)D≤X|ClK[ℓ]|2≪X2−3ℓ+1,ℓ≥5 prime (5.6)∑K=Q(±D)D≤X ClK[3]2≪X23/18,∑K=Q(−D)D≤X ClK[â„“]2≪X2−3â„“+1,ℓ≥5 prime {:(5.6)sum_({:[K=Q(sqrt(+-D))],[D <= X]:})|Cl_(K)[3]|^(2)≪X^(23//18)","quadsum_({:[K=Q(sqrt(-D))],[D <= X]:})|Cl_(K)[â„“]|^(2)≪X^(2-(3)/(â„“+1))","quadâ„“ >= 5" prime ":}\begin{equation*}
\sum_{\substack{K=\mathbb{Q}(\sqrt{ \pm D}) \\ D \leq X}}\left|\mathrm{Cl}_{K}[3]\right|^{2} \ll X^{23 / 18}, \quad \sum_{\substack{K=\mathbb{Q}(\sqrt{-D}) \\ D \leq X}}\left|\mathrm{Cl}_{K}[\ell]\right|^{2} \ll X^{2-\frac{3}{\ell+1}}, \quad \ell \geq 5 \text { prime } \tag{5.6}
\end{equation*}(5.6)∑K=Q(±D)D≤X|ClK[3]|2≪X23/18,∑K=Q(−D)D≤X|ClK[â„“]|2≪X2−3â„“+1,ℓ≥5 primeÂ
as well as results for the kkkkk th moments for all k≥1k≥1k >= 1k \geq 1k≥1. In general, proving tighter control on the size of an exceptional family E0Δ(X)E0Δ(X)E_(0)^(Delta)(X)E_{0}^{\Delta}(X)E0Δ(X) can be used to deduce a better moment bound for |ClK[ℓ]|ClK[â„“]|Cl_(K)[â„“]|\left|\mathrm{Cl}_{K}[\ell]\right||ClK[â„“]|, similar to (5.1). This has recently been exploited by Frei and Widmer, in combination with refinements of the Ellenberg-Venkatesh criterion, to improve moment bounds on ℓâ„“â„“\ellâ„“-torsion for the families of fields studied in [72] (if ℓâ„“â„“\ellâ„“ is sufficiently large); see [41].
Let us mention a connection to elliptic curves; this was after all the setting in which Brumer and Silverman initially posed the ℓâ„“â„“\ellâ„“-torsion Conjecture. Let E(q)E(q)E(q)E(q)E(q) denote the number of isomorphism classes of elliptic curves over QQQ\mathbb{Q}Q with conductor qqqqq. Brumer and Silverman have conjectured that E(q)≪εqεE(q)≪εqεE(q)≪_(epsi)q^(epsi)E(q) \ll_{\varepsilon} q^{\varepsilon}E(q)≪εqε for every q≥1,ε>0q≥1,ε>0q >= 1,epsi > 0q \geq 1, \varepsilon>0q≥1,ε>0 [17]. Conditionally, this follows from GRH combined with a weak form of the Birch-Swinnerton-Dyer conjecture. They also showed this follows from the 3-torsion Conjecture for quadratic fields, by proving
(5.7)E(q)≪εqεmax1≤D≤1728q|ClQ(±D)[3]|, for all ε>0(5.7)E(q)≪εqεmax1≤D≤1728q ClQ(±D)[3], for all ε>0{:(5.7)E(q)≪_(epsi)q^(epsi)max_(1 <= D <= 1728 q)|Cl_(Q(sqrt(+-D)))[3]|","quad" for all "epsi > 0:}\begin{equation*}
E(q) \ll_{\varepsilon} q^{\varepsilon} \max _{1 \leq D \leq 1728 q}\left|\mathrm{Cl}_{\mathbb{Q}(\sqrt{ \pm D})}[3]\right|, \quad \text { for all } \varepsilon>0 \tag{5.7}
\end{equation*}(5.7)E(q)≪εqεmax1≤D≤1728q|ClQ(±D)[3]|, for all ε>0
Duke and Kowalski have combined this with the celebrated asymptotic (5.4) to bound ∑1≤q≤QE(q)≪Q1+ε∑1≤q≤Q E(q)≪Q1+εsum_(1 <= q <= Q)E(q)≪Q^(1+epsi)\sum_{1 \leq q \leq Q} E(q) \ll Q^{1+\varepsilon}∑1≤q≤QE(q)≪Q1+ε for every ε>0ε>0epsi > 0\varepsilon>0ε>0 [30]. (See also [39] for ordering by discriminant.) Pierce, Turnage-Butterbaugh, and Wood have recently proved that for all k≥1k≥1k >= 1k \geq 1k≥1, the kkkkk th moment of 3-torsion in quadratic fields dominates the γkγkgamma k\gamma kγk th moment of E(q)E(q)E(q)E(q)E(q), for a numerical constant γ≈1.9745…γ≈1.9745…gamma~~1.9745 dots\gamma \approx 1.9745 \ldotsγ≈1.9745… coming from [48], which sharpened the relation (5.7). Thus
new moment bounds for E(q)E(q)E(q)E(q)E(q) can be obtained from (5.6), for example. Here is an open problem: prove that ∑1≤q≤QE(q)=o(Q)∑1≤q≤Q E(q)=o(Q)sum_(1 <= q <= Q)E(q)=o(Q)\sum_{1 \leq q \leq Q} E(q)=o(Q)∑1≤q≤QE(q)=o(Q). This would show for the first time that integers that are the conductor of an elliptic curve have density zero in ZZZ\mathbb{Z}Z. In fact, it is conjectured by Watkins that this average is asymptotic to cQ5/6cQ5/6cQ^(5//6)c Q^{5 / 6}cQ5/6 for a certain constant ccccc [94] (building on an analogous conjecture by Brumer-McGuinness for ordering by discriminant [16]).
To conclude, in this section we saw that the truth of the ℓâ„“â„“\ellâ„“-torsion Conjecture is implied by the truth of the well-known Cohen-Lenstra-Martinet heuristics on the distribution of class groups.
5.2. From the perspective of counting number fields of fixed discriminant
Let K/QK/QK//QK / \mathbb{Q}K/Q be a degree nnnnn extension. The Hilbert class field HKHKH_(K)H_{K}HK is the maximal abelian unramified extension of KKKKK, and ClKClKCl_(K)\mathrm{Cl}_{K}ClK is isomorphic to Gal(HK/K)Galâ¡HK/KGal(H_(K)//K)\operatorname{Gal}\left(H_{K} / K\right)Galâ¡(HK/K). A second way to motivate the ℓâ„“â„“\ellâ„“-torsion Conjecture is to count intermediate fields between KKKKK and HKHKH_(K)H_{K}HK.
Here is an argument recorded by Pierce, Turnage-Butterbaugh, and Wood in [73]. Fix a prime ℓâ„“â„“\ellâ„“ and write ClKClKCl_(K)\mathrm{Cl}_{K}ClK additively, so that ClK[ℓ]≃ClK/ℓClKClK[â„“]≃ClK/â„“ClKCl_(K)[â„“]≃Cl_(K)//â„“Cl_(K)\mathrm{Cl}_{K}[\ell] \simeq \mathrm{Cl}_{K} / \ell \mathrm{Cl}_{K}ClK[â„“]≃ClK/â„“ClK. Now define the fixed field L=HKℓClKL=HKâ„“ClKL=H_(K)^(â„“Cl_(K))L=H_{K}^{\ell \mathrm{Cl}_{K}}L=HKâ„“ClK lying between KKKKK and HKHKH_(K)H_{K}HK, so Gal(L/K)≃ClK[ℓ]Galâ¡(L/K)≃ClK[â„“]Gal(L//K)≃Cl_(K)[â„“]\operatorname{Gal}(L / K) \simeq \mathrm{Cl}_{K}[\ell]Galâ¡(L/K)≃ClK[â„“]. Each surjection ClK[ℓ]→Z/ℓZClK[â„“]→Z/â„“ZCl_(K)[â„“]rarrZ//â„“Z\mathrm{Cl}_{K}[\ell] \rightarrow \mathbb{Z} / \ell \mathbb{Z}ClK[â„“]→Z/â„“Z generates an intermediate field MMMMM, with K⊂M⊂LK⊂M⊂LK sub M sub LK \subset M \subset LK⊂M⊂L and deg(M/Q)=nℓdegâ¡(M/Q)=nâ„“deg(M//Q)=nâ„“\operatorname{deg}(M / \mathbb{Q})=n \elldegâ¡(M/Q)=nâ„“. If |ClK[ℓ]|=ℓrClK[â„“]=â„“r|Cl_(K)[â„“]|=â„“^(r)\left|\mathrm{Cl}_{K}[\ell]\right|=\ell^{r}|ClK[â„“]|=â„“r, say, this produces ≈ℓr−1≈ℓr−1~~â„“^(r-1)\approx \ell^{r-1}≈ℓr−1 such fields MMMMM. The crucial point is that since HKHKH_(K)H_{K}HK is an unramified extension, all these fields satisfy a rigid discriminant identity DM=DKℓDM=DKâ„“D_(M)=D_(K)^(â„“)D_{M}=D_{K}^{\ell}DM=DKâ„“. Consequently, if we can count how many number fields of degree nℓnâ„“nâ„“n \ellnâ„“ can share the same fixed discriminant, then we can bound ℓâ„“â„“\ellâ„“-torsion in ClKClKCl_(K)\mathrm{Cl}_{K}ClK. (We have seen this problem before.) We formalize the problem of counting number fields of fixed discriminant as follows:
Property Dn(Δ)Dn(Δ)D_(n)(Delta)\mathbf{D}_{n}(\Delta)Dn(Δ). Fix a degree n≥2n≥2n >= 2n \geq 2n≥2. Property Dn(Δ)Dn(Δ)D_(n)(Delta)\mathbf{D}_{n}(\Delta)Dn(Δ) holds if for every ε>0ε>0epsi > 0\varepsilon>0ε>0 and for every fixed integer D>1D>1D > 1D>1D>1, at most ≪n,εDΔ+ε≪n,εDΔ+ε≪_(n,epsi)D^(Delta+epsi)\ll_{n, \varepsilon} D^{\Delta+\varepsilon}≪n,εDΔ+ε fields K/QK/QK//QK / \mathbb{Q}K/Q of degree nnnnn have DK=DDK=DD_(K)=DD_{K}=DDK=D.
The strategy sketched above ultimately proves that property Dnℓ(Δ)Dnâ„“(Δ)D_(nâ„“)(Delta)\mathbf{D}_{n \ell}(\Delta)Dnâ„“(Δ) implies Cn,ℓ(ℓΔ)Cn,â„“(ℓΔ)C_(n,â„“)(â„“Delta)\mathbf{C}_{n, \ell}(\ell \Delta)Cn,â„“(ℓΔ). This leads inevitably to the question: is property Dnℓ(0)Dnâ„“(0)D_(nâ„“)(0)\mathbf{D}_{n \ell}(0)Dnâ„“(0) true? Here is a conjecture:
Conjecture 5.1 (Discriminant multiplicity conjecture). For each n≥2n≥2n >= 2n \geq 2n≥2, for every ε>0ε>0epsi > 0\varepsilon>0ε>0, and for every integer D>1D>1D > 1D>1D>1, at most ≪n,εDε≪n,εDε≪_(n,epsi)D^(epsi)\ll_{n, \varepsilon} D^{\varepsilon}≪n,εDε fields K/QK/QK//QK / \mathbb{Q}K/Q of degree nnnnn have DK=DDK=DD_(K)=DD_{K}=DDK=D.
This conjecture has been recorded by Duke [29]. It implies the ℓâ„“â„“\ellâ„“-torsion Conjecture, a link noted in [29,35][29,35][29,35][29,35][29,35] and quantified in [73]. Recall the conjecture (3.3) for counting all fields of degree nnnnn and discriminant DK≤XDK≤XD_(K) <= XD_{K} \leq XDK≤X. The Discriminant Multiplicity Conjecture for degree nnnnn would immediately imply Nn(X)≪X1+εNn(X)≪X1+εN_(n)(X)≪X^(1+epsi)N_{n}(X) \ll X^{1+\varepsilon}Nn(X)≪X1+ε, which indicates its level of difficulty. Of course, in general, property Dn(Δ)Dn(Δ)D_(n)(Delta)\mathbf{D}_{n}(\Delta)Dn(Δ) implies Nn(X)≪X1+Δ+εNn(X)≪X1+Δ+εN_(n)(X)≪X^(1+Delta+epsi)N_{n}(X) \ll X^{1+\Delta+\varepsilon}Nn(X)≪X1+Δ+ε for all ε>0ε>0epsi > 0\varepsilon>0ε>0. (In terms of lower bounds, Ellenberg and Venkatesh have noted there can be ≫Dc/loglogD≫Dc/logâ¡logâ¡D≫D^(c//log log D)\gg D^{c / \log \log D}≫Dc/logâ¡logâ¡D extensions K/QK/QK//QK / \mathbb{Q}K/Q with a fixed Galois group and fixed discriminant DDDDD [35].)
The Discriminant Multiplicity Conjecture posits that Dn(0)Dn(0)D_(n)(0)\mathbf{D}_{n}(0)Dn(0) holds for each n≥2n≥2n >= 2n \geq 2n≥2. This is true for n=2n=2n=2n=2n=2, but it is not known for any other degree. For degrees n=3,4,5n=3,4,5n=3,4,5n=3,4,5n=3,4,5, the best-known results currently are D3(1/3)D3(1/3)D_(3)(1//3)\mathbf{D}_{3}(1 / 3)D3(1/3) by [34]; D4(1/2)D4(1/2)D_(4)(1//2)\mathbf{D}_{4}(1 / 2)D4(1/2) as found in [52, 72, 73, 97]; D5(199/200)D5(199/200)D_(5)(199//200)\mathbf{D}_{5}(199 / 200)D5(199/200) as found in [33], building on [9,79]. Currently for n≥6n≥6n >= 6n \geq 6n≥6, the only result for Dn(Δ)Dn(Δ)D_(n)(Delta)\mathbf{D}_{n}(\Delta)Dn(Δ) is a trivial consequence of counting fields of bounded discriminant, as in (3.4), so in particular Δ=c0(logn)2>1Δ=c0(logâ¡n)2>1Delta=c_(0)(log n)^(2) > 1\Delta=c_{0}(\log n)^{2}>1Δ=c0(logâ¡n)2>1 in those cases. It would be very interesting to improve the exponent known for Dn(Δ)Dn(Δ)D_(n)(Delta)\mathbf{D}_{n}(\Delta)Dn(Δ), for any fixed degree n≥3n≥3n >= 3n \geq 3n≥3.
As is the case for many of the problems surveyed in this paper, it can also be profitable to study the problem within a family FFF\mathscr{F}F of degree nnnnn extensions:
Property DF,n(Δ)DF,n(Δ)D_(F,n)(Delta)\mathbf{D}_{\mathscr{F}, n}(\Delta)DF,n(Δ). Fix a degree n≥2n≥2n >= 2n \geq 2n≥2. Property DF,n(Δ)DF,n(Δ)D_(F,n)(Delta)\mathbf{D}_{\mathscr{F}, n}(\Delta)DF,n(Δ) holds iffor every ε>0ε>0epsi > 0\varepsilon>0ε>0 and for every fixed integer D>1D>1D > 1D>1D>1, at most ≪n,εDΔ+ε≪n,εDΔ+ε≪_(n,epsi)D^(Delta+epsi)\ll_{n, \varepsilon} D^{\Delta+\varepsilon}≪n,εDΔ+ε fields K/QK/QK//QK / \mathbb{Q}K/Q in the family FFF\mathscr{F}F have DK=DDK=DD_(K)=DD_{K}=DDK=D.
This is the type of property Pierce, Turnage-Butterbaugh, and Wood used to control the collision problem, in the form (4.11) [72]. Property DF,n(0)DF,n(0)D_(F,n)(0)\mathbf{D}_{\mathscr{F}, n}(0)DF,n(0) has recently been proved by Klüners and Wang, for the family F=Fn(G;X)F=Fn(G;X)F=F_(n)(G;X)\mathscr{F}=\mathscr{F}_{n}(G ; X)F=Fn(G;X) of degree nGnGnGn GnG-extensions for any nilpotent group GGGGG. This was built from the truth of property CF,p(0)CF,p(0)C_(F,p)(0)\mathbf{C}_{\mathscr{F}, p}(0)CF,p(0) for FFF\mathscr{F}F being the family of Galois HHHHH-extensions for HHHHH a ppppp-group, in [54]. There are many other cases where it is an interesting open problem to improve the known bound for PropertyDF,n(Δ)Propertyâ¡DF,n(Δ)Property D_(F,n)(Delta)\operatorname{Property} \mathbf{D}_{\mathscr{F}, n}(\Delta)Propertyâ¡DF,n(Δ).
To conclude, in this section we saw that the ℓâ„“â„“\ellâ„“-torsion Conjecture follows from the Discriminant Multiplicity Conjecture. Now, recall that we saw in the context of bounding ℓâ„“â„“\ellâ„“-torsion that uniform bounds for arbitrarily high moments can imply strong "pointwise" results for every field. Can the method of moments be used to approach the Discriminant Multiplicity Conjecture too? We turn to this idea next.
5.3. From the perspective of counting number fields of bounded discriminant
We come to a third motivation to believe the ℓâ„“â„“\ellâ„“-torsion Conjecture. Recall the definition (3.1) of a family Fn(G;X)Fn(G;X)F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X) of degree nnnnn fields K/QK/QK//QK / \mathbb{Q}K/Q with Gal(K~/Q)Galâ¡(K~/Q)Gal( tilde(K)//Q)\operatorname{Gal}(\tilde{K} / \mathbb{Q})Galâ¡(K~/Q) isomorphic (as a permutation group) to a nontrivial transitive subgroup G⊆SnG⊆SnG subeS_(n)G \subseteq S_{n}G⊆Sn. Each element g∈Gg∈Gg in Gg \in Gg∈G has an index defined by ind(g)=n−ogindâ¡(g)=n−ogind(g)=n-o_(g)\operatorname{ind}(g)=n-o_{g}indâ¡(g)=n−og, where ogogo_(g)o_{g}og is the number of orbits of ggggg when it acts on a set of nnnnn elements. Define a(G)a(G)a(G)a(G)a(G) according to a(G)−1=min{ind(g):1≠g∈G}a(G)−1=min{indâ¡(g):1≠g∈G}a(G)^(-1)=min{ind(g):1!=g in G}a(G)^{-1}=\min \{\operatorname{ind}(g): 1 \neq g \in G\}a(G)−1=min{indâ¡(g):1≠g∈G}; we see that 1n−1≤a(G)≤11n−1≤a(G)≤1(1)/(n-1) <= a(G) <= 1\frac{1}{n-1} \leq a(G) \leq 11n−1≤a(G)≤1. Malle has made a well-known conjecture [65]:
Conjecture 5.2 (Malle). For each n≥2n≥2n >= 2n \geq 2n≥2, for each transitive subgroup G⊆SnG⊆SnG subeS_(n)G \subseteq S_{n}G⊆Sn,
(5.8)|Fn(G;X)|≪G,εXa(G)+ε, for all ε>0(5.8)Fn(G;X)≪G,εXa(G)+ε, for all ε>0{:(5.8)|F_(n)(G;X)|≪_(G,epsi)X^(a(G)+epsi)","quad" for all "epsi > 0:}\begin{equation*}
\left|\mathscr{F}_{n}(G ; X)\right| \ll_{G, \varepsilon} X^{a(G)+\varepsilon}, \quad \text { for all } \varepsilon>0 \tag{5.8}
\end{equation*}(5.8)|Fn(G;X)|≪G,εXa(G)+ε, for all ε>0
The full statement of this conjecture is an open problem. Its difficulty is indicated by the fact that it implies a positive solution to the inverse Galois problem for number fields. (A refinement in [66] specified a power of logXlogâ¡Xlog X\log Xlogâ¡X in place of XεXεX^(epsi)X^{\varepsilon}Xε; counterexamples to this refinement have been found in [50], but the upper bound in (5.8) is expected to be true.)
Malle's Conjecture has been proved for abelian groups, with a strategy by Cohn [24], and asymptotic counts by Mäki [64], Wright [97]. For n=3,4,5n=3,4,5n=3,4,5n=3,4,5n=3,4,5, it is known for SnSnS_(n)S_{n}Sn by the asymptotic (3.3), and for D4D4D_(4)D_{4}D4 by Baily [3] (refined to an asymptotic in [20]). It is known for C2C2C_(2)C_{2}C2 ä¹™ HHHHH under mild conditions on HHHHH (in particular, for at least one group of order nnnnn for every even nnnnn ) by [53], and for Sn×ASn×AS_(n)xx AS_{n} \times ASn×A with AAAAA an abelian group by [67, 90]. For prime
degree pDppDppD_(p)p D_{p}pDp-fields, upper and lower bounds are closely related to ppppp-torsion in class groups of quadratic fields, and have been studied in [23,41,51].
For many groups, it is a difficult open problem to prove upper or lower bounds approaching Malle's prediction. In many results surveyed here, proving a lower bound for |Fn(G;X)|Fn(G;X)|F_(n)(G;X)|\left|\mathscr{F}_{n}(G ; X)\right||Fn(G;X)| has been an important step, to verify a result applies to "almost all" fields in a family. For many groups GGGGG, it is not even known that |Fn(G;X)|≫XβFn(G;X)≫Xβ|F_(n)(G;X)|≫X^(beta)\left|\mathscr{F}_{n}(G ; X)\right| \gg X^{\beta}|Fn(G;X)|≫Xβ for some β>0β>0beta > 0\beta>0β>0 as X→∞X→∞X rarr ooX \rightarrow \inftyX→∞. Here is a tool to prove such a result: suppose f(X,T1,…,Ts)∈Q[X,T1,…,Xs]fX,T1,…,Ts∈QX,T1,…,Xsf(X,T_(1),dots,T_(s))inQ[X,T_(1),dots,X_(s)]f\left(X, T_{1}, \ldots, T_{s}\right) \in \mathbb{Q}\left[X, T_{1}, \ldots, X_{s}\right]f(X,T1,…,Ts)∈Q[X,T1,…,Xs] is a regular polynomial of total degree ddddd in the TiTiT_(i)T_{i}Ti and of degree mmmmm in XXXXX with transitive Galois group G⊂SnG⊂SnG subS_(n)G \subset S_{n}G⊂Sn over Q(T1,…,Ts)QT1,…,TsQ(T_(1),dots,T_(s))\mathbb{Q}\left(T_{1}, \ldots, T_{s}\right)Q(T1,…,Ts). Then |Fn(G;X)|≫f,εXβ−εFn(G;X)≫f,εXβ−ε|F_(n)(G;X)|≫_(f,epsi)X^(beta-epsi)\left|\mathscr{F}_{n}(G ; X)\right| \gg_{f, \varepsilon} X^{\beta-\varepsilon}|Fn(G;X)|≫f,εXβ−ε for every ε>0ε>0epsi > 0\varepsilon>0ε>0, with β=1−|G|−1d(2m−2)β=1−|G|−1d(2m−2)beta=(1-|G|^(-1))/(d(2m-2))\beta=\frac{1-|G|^{-1}}{d(2 m-2)}β=1−|G|−1d(2m−2); this is proved in [72]. For G=AnG=AnG=A_(n)G=A_{n}G=An, a polynomial fffff exhibited by Hilbert can be input to this criterion, implying that |Fn(An;X)|≫Xβn+εFnAn;X≫Xβn+ε|F_(n)(A_(n);X)|≫X^(beta_(n)+epsi)\left|\mathscr{F}_{n}\left(A_{n} ; X\right)\right| \gg X^{\beta_{n}+\varepsilon}|Fn(An;X)|≫Xβn+ε for some βn>0βn>0beta_(n) > 0\beta_{n}>0βn>0, providing the first lower bound that grows like a power of XXXXX. Here is an open problem: for many groups GGGGG, no such polynomial fffff has yet been exhibited.
Now we focus on the conjectured upper bound (5.8) for counting fields with bounded discriminant. For any family F=Fn(G;X)F=Fn(G;X)F=F_(n)(G;X)\mathscr{F}=\mathscr{F}_{n}(G ; X)F=Fn(G;X) of fields, the strong "pointwise" property DF,n(0)DF,n(0)D_(F,n)(0)\mathbf{D}_{\mathscr{F}, n}(0)DF,n(0) implies Malle's "average" upper bound (5.8) for the group GGGGG; see [54]. What is more surprising is that there is a converse to this. This relates to our question: can the method of moments be used to deduce the Discriminant Multiplicity Conjecture? Formally, it can. Given a family FFF\mathscr{F}F of fields, for each integer D≥1D≥1D >= 1D \geq 1D≥1 let m(D)m(D)m(D)m(D)m(D) denote the number of fields K∈FK∈FK inFK \in \mathscr{F}K∈F with DK=DDK=DD_(K)=DD_{K}=DDK=D. If arbitrarily high kkkkk th moment bounds are known for the function m(D)m(D)m(D)m(D)m(D), the Discriminant Multiplicity Conjecture follows; see [73]. But the first moment of m(D)m(D)m(D)m(D)m(D) is the subject of the Malle Conjecture (5.8), so the method of moments certainly seems a difficult avenue to pursue. Yet interestingly, Ellenberg and Venkatesh have shown that in this context the kkkkk th moments can be repackaged as averages.
Informally, the idea is to replace bounding the kkkkk th moment of the function m(D)m(D)m(D)m(D)m(D) for GGGGG-Galois fields in a family FFF\mathscr{F}F by counting fields in a family F(k)F(k)F^((k))\mathscr{F}^{(k)}F(k) of GkGkG^(k)G^{k}Gk-Galois fields. Ellenberg and Venkatesh order the fields in F(k)F(k)F^((k))\mathscr{F}^{(k)}F(k) not by discriminant DKDKD_(K)D_{K}DK, but (roughly speaking) by the square-free kernel DK#DK#D_(K)^(#)D_{K}^{\#}DK# of the discriminant. They generalize the Malle Conjecture to posit that in this ordering, ≪X1+ε≪X1+ε≪X^(1+epsi)\ll X^{1+\varepsilon}≪X1+ε fields in F(k)F(k)F(k)\mathscr{F}(k)F(k) have DK#≤XDK#≤XD_(K)^(#) <= XD_{K}^{\#} \leq XDK#≤X, uniformly for all integers k≥1k≥1k >= 1k \geq 1k≥1. Assuming this conjecture, suppose there are m(D)m(D)m(D)m(D)m(D) many GGGGG-Galois fields K1,…,Km(D)K1,…,Km(D)K_(1),dots,K_(m(D))K_{1}, \ldots, K_{m(D)}K1,…,Km(D) with DKi=DDKi=DD_(K_(i))=DD_{K_{i}}=DDKi=D. Taking composita of kkkkk of these generates at least ≫km(D)k≫km(D)k≫_(k)m(D)^(k)\gg_{k} m(D)^{k}≫km(D)k many GkGkG^(k)G^{k}Gk-Galois fields in the family F(k)F(k)F^((k))\mathscr{F}^{(k)}F(k), with DK#≤DDK#≤DD_(K)^(#) <= DD_{K}^{\#} \leq DDK#≤D. If we suppose m(D)≥Dαm(D)≥Dαm(D) >= D^(alpha)m(D) \geq D^{\alpha}m(D)≥Dα for some α>0α>0alpha > 0\alpha>0α>0 and a sequence of D→∞D→∞D rarr ooD \rightarrow \inftyD→∞, under the generalized Malle Conjecture it must be that αk≤1αk≤1alpha k <= 1\alpha k \leq 1αk≤1 for all k≥1k≥1k >= 1k \geq 1k≥1. Hence ααalpha\alphaα must be arbitrarily small, as desired.
In full generality, Ellenberg and Venkatesh propose a generalized Malle Conjecture in terms of an fffff-discriminant, for any rational class function fffff, and an appropriate generalization aG(f)aG(f)a_(G)(f)a_{G}(f)aG(f) of the exponent in (5.8). They verify that for a particular choice of fffff, this implies the Discriminant Multiplicity Conjecture. More recently, Klüners and Wang have shown directly that Malle's Conjecture (5.8) for all groups GGGGG implies the Discriminant Multiplicity Conjecture (also over any number field) [54].
Let us sum up: the upper bound (5.8) in Malle's Conjecture for all groups GGGGG implies the Discriminant Multiplicity Conjecture. The Discriminant Multiplicity Conjecture implies
the ℓâ„“â„“\ellâ„“-torsion Conjecture. Also, the Discriminant Multiplicity Conjecture for Fn(G;X)Fn(G;X)F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X) (that is, property DF,n(0))DF,n(0){:D_(F,n)(0))\left.\mathbf{D}_{\mathscr{F}, n}(0)\right)DF,n(0)) implies Malle's Conjecture for Fn(G;X)Fn(G;X)F_(n)(G;X)\mathscr{F}_{n}(G ; X)Fn(G;X). Moreover, there is one more converse: Alberts has shown that if the ℓâ„“â„“\ellâ„“-torsion Conjecture is true for all solvable extensions and all primes ℓâ„“â„“\ellâ„“ (even just in an average sense), then Malle's upper bound (5.8) holds for all solvable groups [1]. Thus Malle's Conjecture, the Discriminant Multiplicity Conjecture, and the ℓâ„“â„“\ellâ„“-torsion Conjecture are truly equivalent, when restricted to solvable groups. These relationships provide clear motivation for why so many methods described in this survey have involved counting number fields.
In conclusion, we have seen from three different perspectives that the ℓâ„“â„“\ellâ„“-torsion Conjecture should be true. But as Gauss wrote, "Demonstrationes autem rigorosae harum observationum perdifficiles esse videntur."
FUNDING
This work was partially supported by NSF CAREER DMS-1652173.
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LILLIAN B. PIERCE
Mathematics Department, Duke University, Durham, NC 27708, USA, pierce@ math.duke.edu
POINTS ON SHIMURA VARIETIES MODULO PRIMES
SUG WOO SHIN
ABSTRACT
We survey recent developments on the Langlands-Rapoport conjecture for Shimura varieties modulo primes and an analogous conjecture for Igusa varieties. We discuss resulting implications on the automorphic decomposition of the Hasse-Weil zeta functions and ℓâ„“â„“\ellâ„“ adic cohomology of Shimura varieties, along with further applications to the Langlands correspondence and related problems.
Shimura varieties have been vital to number theory for their intrinsic beauty and wide-ranging applications. They are simultaneously locally symmetric spaces and quasiprojective varieties over number fields, serving as a geometric bridge between automorphic forms and arithmetic. This feature has been particularly fruitful in the Langlands program.
This paper concentrates on the problem of understanding the Hasse-Weil zeta functions and ℓâ„“â„“\ellâ„“-adic cohomology of Shimura varieties following the approach due to Langlands, Kottwitz, Rapoport, and others. As such, we are naturally led to study integral models and special fibers of Shimura varieties at each prime (Section 2), as epitomized by the LanglandsRapoport (LR) conjecture (Section 3). Below is a partial summary of this article:
Central to this paper is Theorem 3.2, asserting that LR0(Sh)LR0(Sh)LR_(0)(Sh)\mathrm{LR}_{0}(\mathrm{Sh})LR0(Sh), a version of the LRLRLR\mathrm{LR}LR conjecture, is true for Shimura varieties of abelian type with good reduction. This is a strengthening of another version LR1(Sh)LR1(Sh)LR_(1)(Sh)\mathrm{LR}_{1}(\mathrm{Sh})LR1(Sh) which was previously verified by Kisin. Even though LR0(Sh)LR0(Sh)LR_(0)(Sh)\mathrm{LR}_{0}(\mathrm{Sh})LR0(Sh) is weaker than the original LR conjecture (still wide open), it opens doors for most applications. Indeed, the diagram shows how LR0(Sh)LR0(Sh)LR_(0)(Sh)\mathrm{LR}_{0}(\mathrm{Sh})LR0(Sh) implies a (stabilized) trace formula for cohomology of Shimura varieties, designated as TF(Sh)TF(Sh)TF(Sh)\mathrm{TF}(\mathrm{Sh})TF(Sh), which in turn leads to interesting applications (Section 4). In Sections 5-6, we survey related problems and directions in the bad reduction case. Finally in Section 7, we review a parallel story for Igusa varieties, where LR0(Sh)LR0(Sh)LR_(0)(Sh)\mathrm{LR}_{0}(\mathrm{Sh})LR0(Sh) provides a key ingredient for proving the analogous assertion LR0(Ig)LR0(Ig)LR_(0)(Ig)\mathrm{LR}_{0}(\mathrm{Ig})LR0(Ig) for Igusa varieties. The dotted vertical arrow suggests that interactions occur between certain applications to Shimura and Igusa varieties, e.g., through Mantovan's formula.
cussed results are valid more generally. In the LR conjecture, we omit ZG(Qp)ZGQpZ_(G)(Q_(p))Z_{G}\left(\mathbb{Q}_{p}\right)ZG(Qp)-equivariance to keep the statements simple. If ΓΓGamma\GammaΓ is a topological group, H(Γ)H(Γ)H(Gamma)\mathscr{H}(\Gamma)H(Γ) is the Hecke algebra of locally constant compactly supported functions on ΓΓGamma\GammaΓ. We write A∞=Z^⊗ZQA∞=Z^⊗ZQA^(oo)= hat(Z)ox_(Z)Q\mathbb{A}^{\infty}=\hat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q}A∞=Z^⊗ZQ for the ring of finite adèles, and A∞,pA∞,pA^(oo,p)\mathbb{A}^{\infty, p}A∞,p for the analogous ring without the ppppp-component. Put Z˘p:=W(F¯p)Z˘p:=WF¯pZ^(˘)_(p):=W( bar(F)_(p))\breve{\mathbb{Z}}_{p}:=W\left(\overline{\mathbb{F}}_{p}\right)Z˘p:=W(F¯p) for the Witt ring of F¯pF¯pbar(F)_(p)\overline{\mathbb{F}}_{p}F¯p, and Q˘p:=Z˘p[1/p]Q˘p:=Z˘p[1/p]Q^(˘)_(p):=Z^(˘)_(p)[1//p]\breve{\mathbb{Q}}_{p}:=\breve{\mathbb{Z}}_{p}[1 / p]Q˘p:=Z˘p[1/p]. Denote by σσsigma\sigmaσ the Frobenius operator on Q˘pQ˘pQ^(˘)_(p)\breve{\mathbb{Q}}_{p}Q˘p or a finite unramified extension of QpQpQ_(p)\mathbb{Q}_{p}Qp. When we have cohomology spaces Hi(X)Hi(X)H^(i)(X)H^{i}(X)Hi(X) (supported on finitely many iiiii 's) with a group action, denote by [H(X)]=∑i≥0(−1)iHi(X)[H(X)]=∑i≥0 (−1)iHi(X)[H(X)]=sum_(i >= 0)(-1)^(i)H^(i)(X)[H(X)]=\sum_{i \geq 0}(-1)^{i} H^{i}(X)[H(X)]=∑i≥0(−1)iHi(X) the alternating sum viewed in a suitable Grothendieck group of representations. For an algebraic group GGGGG over QQQ\mathbb{Q}Q and a field kkkkk over QQQ\mathbb{Q}Q, write Gk:=G×SpecQGk:=G×Specâ¡QG_(k):=Gxx_(Spec Q)G_{k}:=G \times_{\operatorname{Spec} \mathbb{Q}}Gk:=G×Specâ¡Q Spec kkkkk. We quietly fix field
embeddings Q¯↪C,Q¯v↪CQ¯↪C,Q¯v↪Cbar(Q)↪C, bar(Q)_(v)↪C\overline{\mathbb{Q}} \hookrightarrow \mathbb{C}, \overline{\mathbb{Q}}_{v} \hookrightarrow \mathbb{C}Q¯↪C,Q¯v↪C at each place vvvvv of QQQ\mathbb{Q}Q, and identify the residue field of Q¯pQ¯pbar(Q)_(p)\overline{\mathbb{Q}}_{p}Q¯p with F¯pF¯pbar(F)_(p)\overline{\mathbb{F}}_{p}F¯p.
2. SHIMURA VARIETIES WITH GOOD REDUCTION
Let GGGGG be a connected reductive group over QQQ\mathbb{Q}Q, and XXXXX a G(R)G(R)G(R)G(\mathbb{R})G(R)-conjugacy class of RRR\mathbb{R}R-group morphisms ResC/RGm→GRResC/Râ¡Gm→GRRes_(C//R)G_(m)rarrG_(R)\operatorname{Res}_{\mathbb{C} / \mathbb{R}} \mathbb{G}_{m} \rightarrow G_{\mathbb{R}}ResC/Râ¡Gm→GR. We say that (G,X)(G,X)(G,X)(G, X)(G,X) is a Shimura datum if it satisfies axioms (2.1.1.1)-(2.1.1.3) of [10]. Each (G,X)(G,X)(G,X)(G, X)(G,X) determines a conjugacy class of cocharacters μ:Gm→GCμ:Gm→GCmu:G_(m)rarrG_(C)\mu: \mathbb{G}_{m} \rightarrow G_{\mathbb{C}}μ:Gm→GC over CCC\mathbb{C}C, whose field of definition is a number field E=E(G,X)⊂CE=E(G,X)⊂CE=E(G,X)subCE=E(G, X) \subset \mathbb{C}E=E(G,X)⊂C. There is an obvious notion of morphisms between Shimura data.
Thanks to Shimura, Deligne, Borovoi, and Milne, we have a G(A∞)GA∞G(A^(oo))G\left(\mathbb{A}^{\infty}\right)G(A∞)-scheme Sh over EEEEE (in the sense of [10,2.7.1][10,2.7.1][10,2.7.1][10,2.7 .1][10,2.7.1], cf. [31, 1.5.1]), which is a projective limit of quasiprojective varieties over EEEEE with a G(A∞)GA∞G(A^(oo))G\left(\mathbb{A}^{\infty}\right)G(A∞)-action. If (G,X)(G,X)(G,X)(G, X)(G,X) is a Siegel datum, i.e., G=GSp2nG=GSp2nG=GSp_(2n)G=\mathrm{GSp}_{2 n}G=GSp2n and XXXXX is realized by the Siegel half-spaces of genus nnnnn for some n∈Z≥1n∈Z≥1n inZ_( >= 1)n \in \mathbb{Z}_{\geq 1}n∈Z≥1, then we obtain (a projective limit of) Siegel modular varieties as output. There is a hierarchy of Shimura data:
Roughly speaking, Shimura varieties coming from PEL-type data are realized as moduli spaces of abelian varieties with polarizations (P), endomorphisms (E), and level (L) structures. 11^(1){ }^{1}1 This case includes modular curves and, more generally, Siegel modular varieties. A Shimura datum of Hodge type embeds in a Siegel datum by definition, and the corresponding Shimura varieties embed in Siegel modular varieties. Abelian-type data are generalized from those of Hodge type to cover the case when the Dynkin diagram of GQ¯GQ¯G_( bar(Q))G_{\overline{\mathbb{Q}}}GQ¯ consists of only types A, B, C, and D, with a small exception in the type D case, cf. [10, §2.3].
Now we turn to integral models of Shimura varieties in the good reduction case. A starting point is an unramified Shimura datum (G,X,p,E)(G,X,p,E)(G,X,p,E)(G, X, p, \mathscr{E})(G,X,p,E), where (G,X)(G,X)(G,X)(G, X)(G,X) is a Shimura datum, ppppp is a prime, and EEE\mathscr{E}E is a reductive model of GGGGG over ZpZpZ_(p)\mathbb{Z}_{p}Zp. The existence of EEE\mathscr{E}E is equivalent to the condition that GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp is an unramified group (i.e., quasisplit over QpQpQ_(p)\mathbb{Q}_{p}Qp and split over an unramified extension of Qp)Qp{:Q_(p))\left.\mathbb{Q}_{p}\right)Qp). Now we put Kp:=G(Zp)Kp:=GZpK_(p):=G(Z_(p))K_{p}:=\mathcal{G}\left(\mathbb{Z}_{p}\right)Kp:=G(Zp) and consider the G(A∞,p)GA∞,pG(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p)-scheme ShKpShKpSh_(K_(p))\mathrm{Sh}_{K_{p}}ShKp over EEEEE, which is similar to Sh as above but has a fixed level KpKpK_(p)K_{p}Kp at ppppp (while the level subgroup away from ppppp varies). Kisin [28] (p>2)(p>2)(p > 2)(p>2)(p>2) and Kim-Madapusi Pera [27] (p=2)(p=2)(p=2)(p=2)(p=2) proved the following fundamental result.
Theorem 2.1. If (G,X)(G,X)(G,X)(G, X)(G,X) is of abelian type, then there exists a canonical integral model SKpSKpS_(K_(p))\mathscr{S}_{K_{p}}SKp, which is an OE,(p)OE,(p)O_(E,(p))\mathcal{O}_{E,(p)}OE,(p)-scheme with a G(A∞,p)GA∞,pG(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p)-action, such that the generic fiber of SKpSKpS_(K_(p))\mathscr{S}_{K_{p}}SKp is G(A∞,p)GA∞,pG(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p)-equivariantly isomorphic to ShKpShKpSh_(K_(p))\mathrm{Sh}_{K_{p}}ShKp.
Here "canonical" means that SKpSKpS_(K_(p))\mathscr{S}_{K_{p}}SKp is formally smooth over OE,(p)OE,(p)O_(E,(p))\mathcal{O}_{E,(p)}OE,(p) and satisfies the extension property of [28, (2.3.7)], which characterizes SKpSKpS_(K_(p))\mathscr{S}_{K_{p}}SKp uniquely up to a unique isomorphism. The proof of the theorem reduces to the Hodge-type case and utilizes the
1 A caveat is that such a moduli space is in general a finite disjoint union of Shimura varieties due to a possible failure of the Hasse principle for GGGGG. See [34, §8] for details.
known canonical integral models in the Siegel case. Kisin constructs SKpSKpS_(K_(p))\mathscr{S}_{K_{p}}SKp by normalizing the closure of ShKpShKpSh_(K_(p))\mathrm{Sh}_{K_{p}}ShKp in an ambient Siegel modular variety. The key point is to show formal smoothness of SKpSKpS_(K_(p))\mathscr{S}_{K_{p}}SKp over OE,(p)OE,(p)O_(E,(p))\mathcal{O}_{E,(p)}OE,(p) by deformation theory and integral ppppp-adic Hodge theory. The existence of canonical integral models is completely open beyond the abelian-type case.
3. THE LANGLANDS-RAPOPORT CONJECTURE
Given an unramified Shimura datum, the Langlands-Rapoport (LR) conjecture consists of two parts: (i) the existence of canonical integral models and (ii) a group-theoretic description of F¯pF¯pbar(F)_(p)\overline{\mathbb{F}}_{p}F¯p-points of such integral models. We already addressed (i) in Section 2, which is a prerequisite for discussing (ii) in this section. There is an instructive analogy between (ii) and a description of CCC\mathbb{C}C-points [47, $16]. See Section 6.1 below for the case of bad reduction. We recommend the introduction of [31] for a more detailed survey of the content in this section.
3.1. Galois gerbs
Let kkkkk be a perfect field with an algebraic closure k¯k¯bar(k)\bar{k}k¯. A Galois gerb over kkkkk consists of a pair ( G,G)G,G)G,G)G, G)G,G), where GGGGG is a connected linear algebraic group over k¯k¯bar(k)\bar{k}k¯, and GGGGG is a topological group extension (with discrete topology on G(k¯)G(k¯)G( bar(k))G(\bar{k})G(k¯) and profinite topology on Gal(k¯/k)Galâ¡(k¯/k)Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Galâ¡(k¯/k) ),
such that (i) for every g∈Fg∈Fg inFg \in \mathbb{F}g∈F, the conjugation by ggggg on G(k¯)G(k¯)G( bar(k))G(\bar{k})G(k¯) is induced by a k¯k¯bar(k)\bar{k}k¯-group isomorphism π(g)∗G→∼GÏ€(g)∗G→∼Gpi(g)^(**)Grarr"∼"G\pi(g)^{*} G \xrightarrow{\sim} GÏ€(g)∗G→∼G, and (ii) there exists a finite extension K/kK/kK//kK / kK/k in k¯k¯bar(k)\bar{k}k¯ such that πÏ€pi\piÏ€ admits a continuous section over Gal(k¯/K)Galâ¡(k¯/K)Gal( bar(k)//K)\operatorname{Gal}(\bar{k} / K)Galâ¡(k¯/K). If GGGGG is a torus, then (ii) determines a model of GGGGG over kkkkk. We often refer to (G,G)(G,G)(G,G)(G, G)(G,G) as GGGGG and write ↺Δ↺Δ↺Delta\circlearrowleft \Delta↺Δ for GGGGG.
There is a natural notion of morphisms between Galois gerbs over kkkkk. Passing to projective limits, we define pro-Galois gerbs ( G,(F)G,(F)G,(F)G,(F)G,(F) over kkkkk, which still fit in (3.1) but with GGGGG a pro-algebraic group over k¯k¯bar(k)\bar{k}k¯. When GGG\mathbb{G}G is a (pro-)Galois gerb over QQQ\mathbb{Q}Q, we can localize it at each place vvvvv of QQQ\mathbb{Q}Q to obtain a (pro-)Galois gerb over QvQvQ_(v)\mathbb{Q}_{v}Qv, to be denoted by G(v)G(v)G(v)\mathbb{G}(v)G(v).
The most basic example is the neutral Galois gerb FGFGF_(G)\mathscr{F}_{G}FG which arises when GGGGG is already defined over kkkkk. By definition, SG:=G(k¯)⋊Gal(k¯/k)SG:=G(k¯)⋊Galâ¡(k¯/k)S_(G):=G( bar(k))><|Gal( bar(k)//k)\mathbb{S}_{G}:=G(\bar{k}) \rtimes \operatorname{Gal}(\bar{k} / k)SG:=G(k¯)⋊Galâ¡(k¯/k) as a semidirect product with the natural action of Gal(k¯/k)Galâ¡(k¯/k)Gal( bar(k)//k)\operatorname{Gal}(\bar{k} / k)Galâ¡(k¯/k) on G(k)G(k)G(k)G(k)G(k).
We introduce a (pro-)Galois gerb GvGvG_(v)\mathbb{G}_{v}Gv over QvQvQ_(v)\mathbb{Q}_{v}Qv at each vvvvv. Take জ∞জ∞জ_(oo)জ_{\infty}জজ∞ to be the real Weil group (in particular, F∞Δ=Gm,CF∞Δ=Gm,CF_(oo)^(Delta)=G_(m,C)\mathcal{F}_{\infty}^{\Delta}=\mathbb{G}_{m, \mathbb{C}}F∞Δ=Gm,C ); the definition of FpFpF_(p)\mathbb{F}_{p}Fp is involved but intended to encode isocrystals. For v≠p,∞v≠p,∞v!=p,oov \neq p, \inftyv≠p,∞, put Fv:=Gal(Q¯v/Qv)Fv:=Galâ¡Q¯v/QvF_(v):=Gal( bar(Q)_(v)//Q_(v))\mathbb{F}_{v}:=\operatorname{Gal}\left(\overline{\mathbb{Q}}_{v} / \mathbb{Q}_{v}\right)Fv:=Galâ¡(Q¯v/Qv), namely the trivial neutral Galois gerb.
Central to the LR conjecture is a quasimotivic pro-Galois gerb QQQ\mathbb{Q}Q over QQQ\mathbb{Q}Q whose algebraic part QΔQΔQ^(Delta)\mathbb{Q}^{\Delta}QΔ is a pro-torus. The gerb QQQ\mathbb{Q}Q comes equipped with morphisms ζv:Gv→ζv:Gv→zeta_(v):G_(v)rarr\zeta_{v}: \mathscr{G}_{v} \rightarrowζv:Gv→{(v)(v){(v):}\left\{(v)\right.{(v), and the datum ( {,{ζv},ζv{,{zeta_(v)}:}\left\{,\left\{\zeta_{v}\right\}\right.{,{ζv} ) is uniquely characterized up to a suitable equivalence. A quasimotivic gerb (more precisely, its quotient called a pseudomotivic gerb) is devised as a substitute for the Galois gerb which should arise via Tannaka duality from the category of motives over F¯pF¯pbar(F)_(p)\overline{\mathbb{F}}_{p}F¯p. The morphisms ζvζvzeta_(v)\zeta_{v}ζv should come from the fiber functors on the latter
category coming from cohomology and polarization structures. See Langlands-Rapoport [41, §§3-4] (complemented by [56,§8][56,§8][56,§8][56, \S 8]§[56,§8] ) and [58, B2.7, B2.8] for further information.
For each torus TTTTT over QQQ\mathbb{Q}Q and each cocharacter μ:Gm→Tμ:Gm→Tmu:G_(m)rarr T\mu: \mathbb{G}_{m} \rightarrow Tμ:Gm→T (defined over a finite extension of QQQ\mathbb{Q}Q ), there is a recipe [29[29[29[29[29, (3.1.10)] to define a morphism
As a special case, if (T,h)(T,h)(T,h)(T, h)(T,h) is a toral Shimura datum, then we obtain ΨT,μhΨT,μhPsi_(T,mu_(h))\Psi_{T, \mu_{h}}ΨT,μh with μh:Gm→μh:Gm→mu_(h):G_(m)rarr\mu_{h}: \mathbb{G}_{m} \rightarrowμh:Gm→TTTTT coming from hhhhh. In terms of the heuristics for QQQ\mathfrak{Q}Q, the construction of ΨT,μhΨT,μhPsi_(T,mu_(h))\Psi_{T, \mu_{h}}ΨT,μh mirrors the operation of taking the mod ppppp fiber of a CMCMCM\mathrm{CM}CM abelian variety in characteristic 0 .
3.2. Versions of the LRLRLRL RLR conjecture
Let (G,X,p,E)(G,X,p,E)(G,X,p,E)(G, X, p, \mathscr{E})(G,X,p,E) be an unramified Shimura datum. Write ppp\mathfrak{p}p for the prime of EEEEE over ppppp, determined by the field embeddings in Section 1, with residue field k(p)k(p)k(p)k(\mathfrak{p})k(p). A canonical integral model SKpSKpS_(K_(p))\mathscr{S}_{K_{p}}SKp over OEpOEpO_(E_(p))\mathcal{O}_{E_{\mathfrak{p}}}OEp is available in the abelian-type case (Theorem 2.1) and conjectured to exist in general. For the moment, we assume (G,X)(G,X)(G,X)(G, X)(G,X) to be of Hodge type. Then we can take the partition
according to the set III\mathbb{I}I of isogeny classes, and then parametrize the set S(l)S(l)S(l)S(\mathscr{l})S(l) consisting of points in each isogeny class ℓâ„“â„“\ellâ„“ relative to a "base point" of choice in ℓâ„“â„“\ellâ„“. This was obtained by Kisin [29, §1.4], where a subtlety in the notion of ppppp-power isogenies was handled by a result on the connected components of affine Deligne-Lusztig varieties [9]. Each S(χ)S(χ)S(chi)S(\chi)S(χ) is ΦZ×G(A∞,p)ΦZ×GA∞,pPhi^(Z)xx G(A^(oo,p))\Phi^{\mathbb{Z}} \times G\left(\mathbb{A}^{\infty, p}\right)ΦZ×G(A∞,p)-stable, where ΦΦPhi\PhiΦ acts as the geometric Frobenius over k(p)k(p)k(p)k(\mathfrak{p})k(p), and
as a right ΦZ×G(A∞,p)ΦZ×GA∞,pPhi^(Z)xx G(A^(oo,p))\Phi^{\mathbb{Z}} \times G\left(\mathbb{A}^{\infty, p}\right)ΦZ×G(A∞,p)-set, where Xp(ℓ)Xp(â„“)X_(p)(â„“)X_{p}(\mathcal{\ell})Xp(â„“) and Xp(ℓ)Xp(â„“)X^(p)(â„“)X^{p}(\mathcal{\ell})Xp(â„“) account for ppppp-power and prime-to- ppppp isogenies (from a base point). The quotient by Id(Q)Id(Q)I_(d)(Q)I_{d}(\mathbb{Q})Id(Q) takes care of redundant counting up to self-isogenies. Since (G,X)(G,X)(G,X)(G, X)(G,X) is of Hodge type, (3.4) simplifies as Iℓ(Q)∖(Xp(d)×Xp(d))Iâ„“(Q)∖Xp(d)×Xp(d)I_(â„“)(Q)\\(X_(p)(d)xxX^(p)(d))I_{\ell}(\mathbb{Q}) \backslash\left(X_{p}(\mathcal{d}) \times X^{p}(\mathcal{d})\right)Iâ„“(Q)∖(Xp(d)×Xp(d)).
Taking the left quotient by this action (denoted ∖τ∖τ\\_(tau)\backslash_{\tau}∖τ below), we define a ΦZ×G(A∞,p)ΦZ×GA∞,pPhi^(Z)xx G(A^(oo,p))\Phi^{\mathbb{Z}} \times G\left(\mathbb{A}^{\infty, p}\right)ΦZ×G(A∞,p)-set
We just write S(ϕ)S(Ï•)S(phi)S(\phi)S(Ï•) if τÏ„tau\tauÏ„ is trivial. The isomorphism class of Sτ(ϕ)SÏ„(Ï•)S_(tau)(phi)S_{\tau}(\phi)SÏ„(Ï•) depends only on [τ]∈[Ï„]∈[tau]in[\tau] \in[Ï„]∈H(ϕ):=Iϕad (Q)∖Iϕad (A∞)/Iϕ(A∞)H(Ï•):=IÏ•ad (Q)∖IÏ•ad A∞/IÏ•A∞H(phi):=I_(phi)^("ad ")(Q)\\I_(phi)^("ad ")(A^(oo))//I_(phi)(A^(oo))\mathscr{H}(\phi):=I_{\phi}^{\text {ad }}(\mathbb{Q}) \backslash I_{\phi}^{\text {ad }}\left(\mathbb{A}^{\infty}\right) / I_{\phi}\left(\mathbb{A}^{\infty}\right)H(Ï•):=IÏ•ad (Q)∖IÏ•ad (A∞)/IÏ•(A∞) represented by τÏ„tau\tauÏ„. (The right quotient is taken with respect to the multiplication through the natural map Iϕ→Iϕad Iϕ→IÏ•ad I_(phi)rarrI_(phi)^("ad ")I_{\phi} \rightarrow I_{\phi}^{\text {ad }}Iϕ→IÏ•ad .) If ϕ,ϕ′Ï•,ϕ′phi,phi^(')\phi, \phi^{\prime}Ï•,ϕ′ are G(Q¯)G(Q¯)G( bar(Q))G(\overline{\mathbb{Q}})G(Q¯)-conjugate, then Sτ(ϕ)≅Sτ(ϕ′)SÏ„(Ï•)≅Sτϕ′S_(tau)(phi)~=S_(tau)(phi^('))S_{\tau}(\phi) \cong S_{\tau}\left(\phi^{\prime}\right)SÏ„(Ï•)≅SÏ„(ϕ′) and canonically H(ϕ)≅H(ϕ′)H(Ï•)≅Hϕ′H(phi)~=H(phi^('))\mathscr{H}(\phi) \cong \mathscr{H}\left(\phi^{\prime}\right)H(Ï•)≅H(ϕ′). Denoting by JJJ\mathbb{J}J the set of G(Q¯)G(Q¯)G( bar(Q))G(\overline{\mathbb{Q}})G(Q¯) conjugacy classes of admissible morphisms, we write H(L)H(L)H(L)\mathscr{H}(\mathcal{L})H(L) and Sτ(L)SÏ„(L)S_(tau)(L)S_{\tau}(\mathcal{L})SÏ„(L) respectively for H(ϕ)H(Ï•)H(phi)\mathscr{H}(\phi)H(Ï•) and Sτ(ϕ)SÏ„(Ï•)S_(tau)(phi)S_{\tau}(\phi)SÏ„(Ï•), when ggg\mathscr{g}g is the G(Q¯)G(Q¯)G( bar(Q))G(\overline{\mathbb{Q}})G(Q¯)-conjugacy class of ϕÏ•phi\phiÏ•.
It is convenient to name a "rationality" condition on the adelic element τ∈Iϕad (A∞)τ∈IÏ•ad A∞tau inI_(phi)^("ad ")(A^(oo))\tau \in I_{\phi}^{\text {ad }}\left(\mathbb{A}^{\infty}\right)τ∈IÏ•ad (A∞) that is technical but useful. For each maximal torus TTTTT of IϕIÏ•I_(phi)I_{\phi}IÏ• over QQQ\mathbb{Q}Q, we have the maps
where ∂∂del\partial∂ is the connecting homomorphism, and the second map is induced by ZIϕ⊂TZIϕ⊂TZ_(I_(phi))sub TZ_{I_{\phi}} \subset TZIϕ⊂T. We say that τÏ„tau\tauÏ„ is tori-rational if the image of τÏ„tau\tauÏ„ in H1(A∞,T)H1A∞,TH^(1)(A^(oo),T)H^{1}\left(\mathbb{A}^{\infty}, T\right)H1(A∞,T) lies in the subset of the image of H1(Q,T)→H1(A∞,T)H1(Q,T)→H1A∞,TH^(1)(Q,T)rarrH^(1)(A^(oo),T)H^{1}(\mathbb{Q}, T) \rightarrow H^{1}\left(\mathbb{A}^{\infty}, T\right)H1(Q,T)→H1(A∞,T) which maps trivially into the abelianized cohomology of GGGGG, for every TTTTT. This condition depends only on [τ]∈H(ϕ)[Ï„]∈H(Ï•)[tau]inH(phi)[\tau] \in \mathscr{H}(\phi)[Ï„]∈H(Ï•).
We are ready to state versions of the LR conjecture in increasing order of strength. (To be precise, the conjecture requires extra compatibility conditions on τ(J)Ï„(J)tau(J)\tau(\mathcal{J})Ï„(J) under cohomological twistings of ggg\mathscr{g}g in (LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) and (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0), but we avoid mentioning them explicitly in this exposition. See [31, §$2.6-2.7], where these conditions correspond to τ_∈Γ(H)1Ï„_∈Γ(H)1tau _in Gamma(H)_(1)\underline{\tau} \in \Gamma(\mathscr{H})_{1}Ï„_∈Γ(H)1 and τ_∈Γ(H)0Ï„_∈Γ(H)0tau _in Gamma(H)_(0)\underline{\tau} \in \Gamma(\mathscr{H})_{0}Ï„_∈Γ(H)0, respectively. With this correction, the Langlands-Rapoport- τÏ„tau\tauÏ„ conjecture therein is exactly (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) below.)
Conjecture 3.1. The following assertions hold true:
(LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) There exists a ΦZ×G(A∞,p)ΦZ×GA∞,pPhi^(Z)xx G(A^(oo,p))\Phi^{\mathbb{Z}} \times G\left(\mathbb{A}^{\infty, p}\right)ΦZ×G(A∞,p)-equivariant bijection
for some family of elements {τ(J)∈H(d)}L∈J{Ï„(J)∈H(d)}L∈J{tau(J)inH(d)}_(LinJ)\{\tau(\mathcal{J}) \in \mathscr{H}(\mathcal{d})\}_{\mathcal{L} \in \mathbb{J}}{Ï„(J)∈H(d)}L∈J.
(LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) The conclusion of (LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) holds with τ(H)Ï„(H)tau(H)\tau(\mathcal{H})Ï„(H) tori-rational for every G∈JG∈JGinJ\mathcal{G} \in \mathbb{J}G∈J.
(LR) The conclusion of (LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) holds with τ(H)Ï„(H)tau(H)\tau(\mathcal{H})Ï„(H) trivial for every J∈JJ∈JJinJ\mathcal{J} \in \mathbb{J}J∈J.
Statement (LR) is nothing but the original LR conjecture. In the Hodge-type case (to which the abelian-type case can be reduced in practice), a natural approach in view of (3.3) is to establish a bijection J∈I↔L∈JJ∈I↔L∈JJinIharrLinJ\mathcal{J} \in \mathbb{I} \leftrightarrow \mathcal{L} \in \mathbb{J}J∈I↔L∈J such that there exists a ΦZ×G(A∞,p)ΦZ×GA∞,pPhi^(Z)xx G(A^(oo,p))\Phi^{\mathbb{Z}} \times G\left(\mathbb{A}^{\infty, p}\right)ΦZ×G(A∞,p)-equivariant bijection S(ℓ)≅Sτ(J)(J)S(â„“)≅SÏ„(J)(J)S(â„“)~=S_(tau(J))(J)S(\mathcal{\ell}) \cong S_{\tau(\mathcal{J})}(\mathcal{J})S(â„“)≅SÏ„(J)(J) with constraints on τ(J)Ï„(J)tau(J)\tau(\mathcal{J})Ï„(J) as in the conjecture.
It is known ([31, §3], cf. [42, 46]) that (LR) implies (3.7) below, which is the gateway to applications, but (LR) remains to be completely open even for Siegel modular varieties (of genus ≥2≥2>= 2\geq 2≥2 ). Milne proved that the original LR conjecture follows from the Hodge conjecture for CMCMCM\mathrm{CM}CM abelian varieties (see [48, P. 4] and the references therein), but the latter conjecture is also wide open.
On a positive note, Kisin [29] made a major breakthrough to prove (LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) for all unramified Shimura data (G,X,p,E)(G,X,p,E)(G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) of abelian type. Unfortunately, (LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) by itself is not strong enough for the next steps. This motivated us to formulate and prove the strengthening (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) in [31], which suffices for the trace formula (Section 3.3) and applications (Section 4) below.
Theorem 3.2. For every unramified Shimura datum (G,X,p,E)(G,X,p,E)(G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) of abelian type, Conjecture (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) holds true.
Let us sketch some ideas of proof when (G,X)(G,X)(G,X)(G, X)(G,X) is of Hodge type. The reduction to this case is nontrivial and convoluted, cf. [31, $6]. Already in [29], Kisin proved a refinement of (LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) in order to propagate (LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) through Deligne's formalism of connected Shimura varieties. With that said, we focus on the Hodge-type setting for simplicity.
The proof consists of two parts: (i) constructing a bijection d∈I↔d∈Jd∈I↔d∈JdinIharrdinJ\mathscr{d} \in \mathbb{I} \leftrightarrow \mathscr{d} \in \mathbb{J}d∈I↔d∈J and (ii) showing that S(d)≅Sτ(J)(d)S(d)≅SÏ„(J)(d)S(d)~=S_(tau(J))(d)S(\mathcal{d}) \cong S_{\tau(\mathcal{J})}(\mathcal{d})S(d)≅SÏ„(J)(d) with some control over τ(d)Ï„(d)tau(d)\tau(\mathcal{d})Ï„(d). A crucial idea is to use special point data, namely toral Shimura data (T,hTT,hT(T,h_(T):}\left(T, h_{T}\right.(T,hT ) with embeddings into (G,X)(G,X)(G,X)(G, X)(G,X), to probe both sides of the bijection. Such data can be mapped into III\mathbb{I}I by taking mod ppppp of the corresponding special points on ShKpShKpSh_(K_(p))\mathrm{Sh}_{K_{p}}ShKp, and to JJJ\mathbb{J}J by composing (3.2) with the induced embedding FT↪FGFT↪FGF_(T)↪F_(G)\mathfrak{F}_{T} \hookrightarrow \mathfrak{F}_{G}FT↪FG. The map to III\mathbb{I}I is onto by Kisin [29], generalizing Honda's result on CMCMCM\mathrm{CM}CM lifting of an abelian variety over F¯pF¯pbar(F)_(p)\overline{\mathbb{F}}_{p}F¯p up to isogeny. The surjectivity onto JJJ\mathbb{J}J is due to LanglandsRapoport [41]:
From each of III\mathbb{I}I and JJJ\mathbb{J}J, Kisin [29] constructed Kottwitz triples consisting of certain conjugacy classes on GGGGG up to an equivalence, and showed that the outer diagram above commutes. This determines a bijection I≅JI≅JI~=J\mathbb{I} \cong \mathbb{J}I≅J up to a finite ambiguity since the maps to Kottwitz triples have finite fibers. However, S(d)S(d)S(d)S(\mathscr{d})S(d) and S(F)S(F)S(F)S(\mathcal{F})S(F) need not be isomorphic through this bijection, since Kottwitz triples forget part of their structures. The possible deviation is recorded by τ(J)Ï„(J)tau(J)\tau(\mathscr{J})Ï„(J), which is a priori under little control. This is still enough for deducing (LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1).
3.3. From the LR conjecture to a stabilized trace formula
Here we return to a general unramified Shimura datum (G,X,p,E)(G,X,p,E)(G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) (possibly not of abelian type). Set r∈Zr∈Zr inZr \in \mathbb{Z}r∈Z to be the inertia degree of ppp\mathfrak{p}p over ppppp. As indicated above, ( LR0)LR0{:LR_(0))\left.\mathrm{LR}_{0}\right)LR0)
is designed as a substitute for (LR) to imply the following formula predicted by [32,41][32,41][32,41][32,41][32,41]. The implication is shown in [31, §3] (refer to the latter for undefined notation):
Here ϕ(j)Ï•(j)phi^((j))\phi^{(j)}Ï•(j) is an explicit function in the unramified Hecke algebra of G(Qpjr)GQpjrG(Q_(p)^(jr))G\left(\mathbb{Q}_{p}{ }^{j r}\right)G(Qpjr) (with respect to G(Zpjr)GZpjrG(Z_(p^(jr)))\mathcal{G}\left(\mathbb{Z}_{p^{j r}}\right)G(Zpjr) ), and the sum runs over certain group-theoretic data c (called Kottwitz parameters) fibered over the set of stable conjugacy classes in G(Q)G(Q)G(Q)G(\mathbb{Q})G(Q) which are elliptic in G(R)G(R)G(R)G(\mathbb{R})G(R). Here c determines an explicit constant c(c)∈Q,γ(c)∈G(A∞,p)c(c)∈Q,γ(c)∈GA∞,pc(c)inQ,gamma(c)in G(A^(oo,p))c(\mathfrak{c}) \in \mathbb{Q}, \gamma(\mathfrak{c}) \in G\left(\mathbb{A}^{\infty, p}\right)c(c)∈Q,γ(c)∈G(A∞,p) up to conjugacy, and δ(c)∈δ(c)∈delta(c)in\delta(\mathrm{c}) \inδ(c)∈G(Qpjr)GQpjrG(Q_(p^(jr)))G\left(\mathbb{Q}_{p^{j r}}\right)G(Qpjr) up to σσsigma\sigmaσ-conjugacy. In particular, the orbital integral Oγ(c)(f∞,p)Oγ(c)f∞,pO_(gamma(c))(f^(oo,p))O_{\gamma(\mathfrak{c})}\left(f^{\infty, p}\right)Oγ(c)(f∞,p) on G(A∞,p)GA∞,pG(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p) and the σσsigma\sigmaσ-twisted orbital integral TOδ(c)(ϕ(j))TOδ(c)Ï•(j)TO_(delta(c))(phi^((j)))T O_{\delta(c)}\left(\phi^{(j)}\right)TOδ(c)(Ï•(j)) on G(Qpjr)GQpjrG(Q_(p^(jr)))G\left(\mathbb{Q}_{p^{j r}}\right)G(Qpjr) are well defined. Stabilizing the right-hand side, we arrive at the following, which is a rough version of [31, THM. 3 AND 4].
Theorem 3.3. Assume that (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) is true. For every f∞,p∈H(G(A∞,p))f∞,p∈HGA∞,pf^(oo,p)inH(G(A^(oo,p)))f^{\infty, p} \in \mathscr{H}\left(G\left(\mathbb{A}{ }^{\infty, p}\right)\right)f∞,p∈H(G(A∞,p)), there exists a constant j0j0j_(0)j_{0}j0 such that for every j≥j0j≥j0j >= j_(0)j \geq j_{0}j≥j0, a formula of the following form holds.
where Eell(G)Eell(G)E_(ell)(G)\mathcal{E}_{\mathrm{ell}}(G)Eell(G) is the set of elliptic endoscopic data for GGGGG up to isomorphism, and STelleSTelleST_(ell)^(e)\mathrm{ST}_{\mathrm{ell}}^{e}STelle is the stable elliptic distribution associated with the endoscopic datum eeeee.
In light of Theorem 3.2, the conclusion of the theorem is unconditionally true for (G,X)(G,X)(G,X)(G, X)(G,X) of abelian type. We can easily allow a nonconstant coefficient as done in [31].
The proof of (3.7) from (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) is mostly close to the deduction from (LR) (cf. [46]), and starts from the fixed-point formula for (improper) varieties over finite fields due to Fujiwara and Varshavsky [13,70]; this explains the condition on jjjjj. To compute the cohomology of the generic fiber via that of the special fiber, we apply Lan-Stroh's result [39]. Tori-rationality in (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) is the main point to ensure that the fixed-point counting is not affected by the presence of τ(F)Ï„(F)tau(F)\tau(\mathcal{F})Ï„(F) even if τ(L)Ï„(L)tau(L)\tau(\mathcal{L})Ï„(L) is nontrivial. The stabilization from (3.7) to Theorem 3.3 follows the argument in [32] with small improvements to work without technical hypotheses. We note that fe,∞,pfe,∞,pf^(e,oo,p)f^{e, \infty, p}fe,∞,p is the Langlands-Shelstad transfer of f∞,pf∞,pf^(oo,p)f^{\infty, p}f∞,p whereas fpe,(j)fpe,(j)f_(p)^(e,(j))f_{p}^{e,(j)}fpe,(j) and f∞ef∞ef_(oo)^(e)f_{\infty}^{e}f∞e are constructed differently. (See [31, §8.2].) As usual in endoscopy, auxiliary zzzzz-extensions are chosen if the derived subgroup of GGGGG is not simply connected, and the right-hand side of (3.8) should be interpreted appropriately.
Remark 3.4. Sometimes it is possible to obtain (3.8) bypassing any version of the LR conjecture. When (G,X)(G,X)(G,X)(G, X)(G,X) is of PEL type A or C, this is done by Kottwitz [34]; for Hodge-type data, this is worked out by Lee [42]. It is unclear how their methods interact with connected components of Shimura varieties, so their results do not easily extend to the abelian-type setup. In contrast, the formalism of the LR conjecture is well suited to such extensions.
Remark 3.5. If the adjoint quotient G/ZGG/ZGG//Z_(G)G / Z_{G}G/ZG is isotropic over QQQ\mathbb{Q}Q or, equivalently, if Sh is not proper over EEEEE (at each fixed level), it is desirable to prove the analogue of (3.8) for the intersection cohomology of the Satake-Baily-Borel compactification; see [51, §84-5] for what
new problems need to be solved. This has been carried out for certain unitary and orthogonal Shimura varieties, as well as Siegel modular varieties, in [50,52,73][50,52,73][50,52,73][50,52,73][50,52,73].
4. APPLICATIONS
4.1. The Hasse-Weil zeta functions and ℓâ„“â„“\ellâ„“-adic cohomology
As pioneered by Eichler, Shimura, Deligne, Kuga, Sato, and Ihara, a central problem on Shimura varieties is to compute their ζζzeta\zetaζ-functions and ℓâ„“â„“\ellâ„“-adic cohomology. The goals are (i) to express the ζζzeta\zetaζ-function as a quotient of products of automorphic LLLLL-functions (thereby deduce a meromorphic continuation and a functional equation when the LLLLL-functions are sufficiently understood), cf. [6, coNJ. 5.2], and (ii) to decompose the ℓâ„“â„“\ellâ„“-adic cohomology according to automorphic representations and identify the Galois action on each piece. To this end, Langlands and Kottwitz developed a robust method in a series of papers in the 1970-1980s (from [40] to [32]). At the heart is a comparison between the Arthur-Selberg trace formula and a conjectural trace formula for the Hecke-Frobenius action on the cohomology at good primes ppppp, where the latter should come from a fixed-point formula for the special fiber of Shimura varieties modulo ppppp.
When G/ZGG/ZGG//Z_(G)G / Z_{G}G/ZG is anisotropic over QQQ\mathbb{Q}Q (equivalently, when Sh is an inverse limit of projective varieties), Theorem 3.3 should be sufficient for the goals (i) and (ii) (up to semisimplifying the Galois action), by following the outline in [32, 888−10]888−10]888-10]888-10]888−10]. We say "should" for two reasons. Firstly, we do not have enough knowledge about automorphic representations in general (e.g., endoscopic classification, cf. [32, §8]). Thus complete details have not been worked out apart from low-rank examples, some special cases such as [33], or under simplifying hypotheses. Secondly, we typically need a positive answer to the following problem to proceed. 22^(2){ }^{2}2 The reason is that STeSTeST^(e)\mathrm{ST}^{e}STe should admit a relatively clean spectral expansion in terms of the discrete automorphic spectrum of endoscopic groups for GGGGG, but the spectral interpretation of STell eSTell eST_("ell ")^(e)\mathrm{ST}_{\text {ell }}^{e}STell e is expected to be quite complicated in general.
Problem 4.1. Assume that G/ZGG/ZGG//Z_(G)G / Z_{G}G/ZG is QQQ\mathbb{Q}Q-anisotropic. In (3.8), prove that
STelle(fe,∞,pfpe,(j)f∞e)=STe(fe,∞,pfpe,(j)f∞e),∀e∈Eell(G)STelleâ¡fe,∞,pfpe,(j)f∞e=STeâ¡fe,∞,pfpe,(j)f∞e,∀e∈Eell(G)ST_(ell)^(e)(f^(e,oo,p)f_(p)^(e,(j))f_(oo)^(e))=ST^(e)(f^(e,oo,p)f_(p)^(e,(j))f_(oo)^(e)),quad AA e inE_(ell)(G)\operatorname{ST}_{\mathrm{ell}}^{e}\left(f^{e, \infty, p} f_{p}^{e,(j)} f_{\infty}^{e}\right)=\operatorname{ST}^{e}\left(f^{e, \infty, p} f_{p}^{e,(j)} f_{\infty}^{e}\right), \quad \forall e \in \mathcal{E}_{\mathrm{ell}}(G)STelleâ¡(fe,∞,pfpe,(j)f∞e)=STeâ¡(fe,∞,pfpe,(j)f∞e),∀e∈Eell(G)
where STeSTeST^(e)\mathrm{ST}^{e}STe stands for the stable distribution as defined in [52,$5.4][52,$5.4][52,$5.4][52, \$ 5.4][52,$5.4].
Although this problem is open, there are quite a few examples where it is known either by the nature of GGGGG or under a simplifying hypothesis on the test function. This provides a starting point for the Langlands correspondence (Section 4.2 below).
Now we remove the assumption on G/ZGG/ZGG//Z_(G)G / Z_{G}G/ZG. In fact, the argument outlined in [32, $88−10]$88−10]$88-10]\$ 8 \mathbf{8 - 1 0 ]}$88−10] is given in this generality, conditional on an affirmative answer to the following.
2 A shortcut getting around Problem 4.1 is possible when GGGGG has "no endoscopy," e.g., if GGGGG is a form of GL2GL2GL_(2)\mathrm{GL}_{2}GL2 or a certain unitary similitude group as in [33].
where IH(Sh¯,Q¯l)IHSh¯,Q¯lIH( bar(Sh), bar(Q)_(l))\mathrm{IH}\left(\overline{\mathrm{Sh}}, \overline{\mathbb{Q}}_{l}\right)IH(Sh¯,Q¯l) is the intersection cohomology of the Satake-Baily-Borel compactification of Sh (see [51, 3.4-3.5], for instance).
To obtain (4.1) from Theorem 3.3, one has to match the nonelliptic terms in STeSTeST^(e)\mathrm{ST}^{e}STe with the contribution to [IH(Sh¯,Q¯l)]IHâ¡Sh¯,Q¯l[IH( bar(Sh), bar(Q)_(l))]\left[\operatorname{IH}\left(\overline{\mathrm{Sh}}, \overline{\mathbb{Q}}_{l}\right)\right][IHâ¡(Sh¯,Q¯l)] from the boundaries. As a special case, if G/ZGG/ZGG//Z_(G)G / Z_{G}G/ZG is QQQ\mathbb{Q}Q anisotropic, then IHi(Sh¯,Q¯l)=Hci(Sh,Q¯l)IHiSh¯,Q¯l=HciSh,Q¯lIH^(i)( bar(Sh), bar(Q)_(l))=H_(c)^(i)(Sh, bar(Q)_(l))\mathrm{IH}^{i}\left(\overline{\mathrm{Sh}}, \overline{\mathbb{Q}}_{l}\right)=H_{c}^{i}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)IHi(Sh¯,Q¯l)=Hci(Sh,Q¯l) for each i≥0i≥0i >= 0i \geq 0i≥0, and the identity of Problem 4.1 should hold since there are no boundaries. In this sense, Problem 4.2 generalizes Problem 4.1. Problem 4.2 has been solved for Siegel modular varieties and certain unitary/orthogonal Shimura varieties by Morel and Zhu [50,52,73][50,52,73][50,52,73][50,52,73][50,52,73].
4.2. The global Langlands correspondence
The computation of ℓâ„“â„“\ellâ„“-adic cohomology in Section 4.1 often leads to new instances of the global Langlands correspondence satisfying a local-global compatibility in the direction from automorphic representations to Galois representations, roughly stated as follows. Refer to Buzzard-Gee [7] for the definition of LLLLL-algebraicity and a full discussion of the conjecture, including a variant conjecture for CCCCC-algebraic representations.
Conjecture 4.3. Let FFFFF be a number field, and π=⊗v′πvÏ€=⊗v′πvpi=ox_(v)^(')pi_(v)\pi=\otimes_{v}^{\prime} \pi_{v}Ï€=⊗v′πv an L-algebraic cuspidal automorphic representation of G(AF)GAFG(A_(F))G\left(\mathbb{A}_{F}\right)G(AF). Then for each prime ℓâ„“â„“\ellâ„“ and each isomorphism ι:Q¯l≅Cι:Q¯l≅Ciota: bar(Q)_(l)~=C\iota: \overline{\mathbb{Q}}_{l} \cong \mathbb{C}ι:Q¯l≅C, there exists a continuous representation ρℓ,l:Gal(F¯/F)→LG(Q¯l)Ïâ„“,l:Galâ¡(F¯/F)→LGQ¯lrho_(â„“,l):Gal( bar(F)//F)rarr^(L)G( bar(Q)_(l))\rho_{\ell, l}: \operatorname{Gal}(\bar{F} / F) \rightarrow{ }^{L} G\left(\overline{\mathbb{Q}}_{l}\right)Ïâ„“,l:Galâ¡(F¯/F)→LG(Q¯l) such that the restriction of ρℓ,ιÏâ„“,ιrho_(â„“,iota)\rho_{\ell, \iota}Ïâ„“,ι to Gal(F¯v/Fv)Galâ¡F¯v/FvGal( bar(F)_(v)//F_(v))\operatorname{Gal}\left(\bar{F}_{v} / F_{v}\right)Galâ¡(F¯v/Fv) is isomorphic to the unramified Langlands parameter of πvÏ€vpi_(v)\pi_{v}Ï€v at almost all finite places vvvvv of FFFFF (where πvÏ€vpi_(v)\pi_{v}Ï€v is unramified).
The relevance of Shimura varieties to the conjecture is as follows. A Shimura datum (G,X)(G,X)(G,X)(G, X)(G,X) determines a representation rX:LG→GL(V)rX:LG→GL(V)r_(X):^(L)G rarrGL(V)r_{X}:{ }^{L} G \rightarrow \mathrm{GL}(V)rX:LG→GL(V) (up to isomorphism). Then one expects that the Galois representation rX∘ρℓ,LrX∘Ïâ„“,Lr_(X)@rho_(â„“,L)r_{X} \circ \rho_{\ell, L}rX∘Ïâ„“,L is realized in the ℓâ„“â„“\ellâ„“-adic cohomology of the associated Shimura varieties (more precisely, the π∞π∞pi^(oo)\pi^{\infty}π∞-isotypic part thereof), with several caveats including normalization issues (e.g., CCCCC-algebraic vs LLLLL-algebraic), Arthur packets, and endoscopic problems. These caveats often present much difficulty, and even if they are ignored, it is generally a subtle group-theoretic problem to recover ρℓ,ιÏâ„“,ιrho_(â„“,iota)\rho_{\ell, \iota}Ïâ„“,ι from r∘ρℓ,lr∘Ïâ„“,lr@rho_(â„“,l)r \circ \rho_{\ell, l}r∘Ïâ„“,l for a set of representations rrrrr of LGLG^(L)G{ }^{L} GLG. (Over global function fields, V. Lafforgue [38] solved the analogous problem in a revolutionary way via generalized pseudocharacters.)
The most fundamental case of Conjecture 4.3 is when G=GLnG=GLnG=GL_(n)G=\mathrm{GL}_{n}G=GLn. When FFFFF is a totally real or CM field and πÏ€pi\piÏ€ satisfies a suitable self-duality condition, then the conjecture is proven in a series of papers making use of PEL-type Shimura varieties arising from a unitary similitude group by Clozel, Kottwitz, and others. (See [67] for a discussion and references.) The duality condition allows πÏ€pi\piÏ€ to "descend" to an automorphic representation on the unitary similitude group as first observed by Clozel. The self-duality condition was later removed independently by Harris-Lan-Taylor-Thorne and Scholze [20,61], by exquisite ppppp-adic congruences which are beyond the scope of this article.
The above results for GLnGLnGL_(n)\mathrm{GL}_{n}GLn imply new cases of Conjecture 4.3 (or a weaker form) for quasisplit unitary, symplectic, or special orthogonal groups GGGGG over a totally real or CMCMCM\mathrm{CM}CM field via twisted endoscopy by Arthur and Mok [1,49]. However, the conjecture for symplectic or orthogonal similitude groups does not follow easily. To get a feel for the difference, note that the dual groups of Sp2n,SO2nSp2n,SO2nSp_(2n),SO_(2n)\mathrm{Sp}_{2 n}, \mathrm{SO}_{2 n}Sp2n,SO2n are SO2n+1,SO2nSO2n+1,SO2nSO_(2n+1),SO_(2n)\mathrm{SO}_{2 n+1}, \mathrm{SO}_{2 n}SO2n+1,SO2n, which are embeddable in GL2n+1,GL2nGL2n+1,GL2nGL_(2n+1),GL_(2n)\mathrm{GL}_{2 n+1}, \mathrm{GL}_{2 n}GL2n+1,GL2n. In contrast, the dual groups of GSp2nGSp2nGSp_(2n)\mathrm{GSp}_{2 n}GSp2n and GSO2nGSO2nGSO_(2n)\mathrm{GSO}_{2 n}GSO2n are GSpin 2n+12n+12n+12 n+12n+1 and GSpin2nGSpin2nGSpin_(2n)\mathrm{GSpin}_{2 n}GSpin2n, whose faithful representations have dimensions at least 2n2n2^(n)2^{n}2n (achieved by the spin representation). In [35,37], Conjecture 4.3 is verified for GSp2nGSp2nGSp_(2n)\mathrm{GSp}_{2 n}GSp2n and a (possibly outer) form of GSO2nGSO2nGSO_(2n)\mathrm{GSO}_{2 n}GSO2n over a totally real field FFFFF under a simplifying hypothesis on πÏ€pi\piÏ€. The basic input comes from Shimura varieties of abelian type associated with a form of ResF/QGSp2nResF/Qâ¡GSp2nRes_(F//Q)GSp_(2n)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GSp}_{2 n}ResF/Qâ¡GSp2n, resp. ResF/QGSO2nResF/Qâ¡GSO2nRes_(F//Q)GSO_(2n)\operatorname{Res}_{F / \mathbb{Q}} \mathrm{GSO}_{2 n}ResF/Qâ¡GSO2n, where rXrXr_(X)r_{X}rX is essentially the spin representation, resp. a half-spin representation. Both problems in Section 4.1 have positive answers in that setup.
5. SHIMURA VARIETIES WITH BAD REDUCTION, PART I
Let (G,X)(G,X)(G,X)(G, X)(G,X) be a Shimura datum. At each prime ppppp such that (G,X)(G,X)(G,X)(G, X)(G,X) can be promoted to an unramified Shimura datum (G,X,p,E)(G,X,p,E)(G,X,p,E)(G, X, p, \mathscr{E})(G,X,p,E), we discuss three methods to study the cohomology Hc(Sh,Q¯l)HcSh,Q¯lH_(c)(Sh, bar(Q)_(l))H_{c}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)Hc(Sh,Q¯l) as a G(A∞)×Gal(E¯p/Ep)GA∞×Galâ¡E¯p/EpG(A^(oo))xx Gal( bar(E)_(p)//E_(p))G\left(\mathbb{A}^{\infty}\right) \times \operatorname{Gal}\left(\bar{E}_{\mathfrak{p}} / E_{\mathfrak{p}}\right)G(A∞)×Galâ¡(E¯p/Ep)-module at each prime ppp\mathfrak{p}p of EEEEE over ppppp. 33^(3){ }^{3}3 The "bad reduction" in the section title means that the level subgroup Kp′⊂Kp=E(Zp)Kp′⊂Kp=EZpK_(p)^(')subK_(p)=E(Z_(p))K_{p}^{\prime} \subset K_{p}=\mathscr{E}\left(\mathbb{Z}_{p}\right)Kp′⊂Kp=E(Zp) at ppppp is allowed to be arbitrarily small, in which case integral models typically have bad reduction mod ppppp. The complicated geometry may be understood better through stratifications.
There are several stratifications of interest on Shimura varieties (cf. [22]) but the most relevant to us is the Newton stratification. In the Hodge-type case, this yields a partition of the modpmodpmod p\bmod pmodp Shimura variety into finitely many locally closed subsets, which can be equipped with the reduced subscheme structure, according to the isogeny class of ppppp-divisible groups with additional structure. The unique closed stratum is called the basic stratum and corresponds to the ppppp-divisible group that is "most supersingular" under the given constraint.
The first method is a ppppp-adic uniformization of Shimura varieties as pioneered by ÄŒerednik and Drinfeld, and further developed by Rapoport-Zink, Fargues, Kim, and Howard-Pappas [11,23,26,57]. Let ShKpKp′basic ShKpKp′basic Sh_(K^(p)K_(p)^('))^("basic ")\mathrm{Sh}_{K^{p} K_{p}^{\prime}}^{\text {basic }}ShKpKp′basic denote the basic locus in the rigid analytification of ShKpKp′ShKpKp′Sh_(K^(p)K_(p)^('))\mathrm{Sh}_{K^{p} K_{p}^{\prime}}ShKpKp′ over EpEpE_(p)E_{\mathfrak{p}}Ep, defined to be the preimage of the basic stratum under the specialization map towards the special fiber. The fundamental result asserts that ShKpKp′basi ShKpKp′basi Sh_(K^(p)K_(p)^('))^("basi ")\mathrm{Sh}_{K^{p} K_{p}^{\prime}}^{\text {basi }}ShKpKp′basi is uniformized by the Rapoport-Zink space with level Kp′Kp′K_(p)^(')K_{p}^{\prime}Kp′ arising from the corresponding basic isogeny class. A prominent application is to prove new cases of the Kottwitz conjecture on the cohomology of basic Rapoport-Zink spaces and their generalizations [11,19,25]. Hansen's work [19] points to a synergy between the global method here and Fargues-Scholze's purely local geometric construction of the local Langlands correspondence [12].
Next we discuss the Harris-Taylor method [21, cHAPS. Iv-v] based on a product structure, namely coverings of Newton strata by the products of Igusa varieties and RapoportZink spaces. The outcome is known as Mantovan's formula [44] (generalizing [21, cHAP. Iv]), which expresses the cohomology of Newton strata in terms of that of Igusa varieties and Rapoport-Zink spaces. In the basic case, this is closely related to the ppppp-adic uniformization. Hamacher-Kim [18] extended Mantovan's formula and the product structure to Hodge-type Shimura varieties.
To go further, it is desirable to understand the cohomology of Igusa varieties - we address this problem in Section 7 below. Granting this, and putting different Newton strata together, we have a formula relating [Hc(Sh,Q¯l)]HcSh,Q¯l[H_(c)(Sh, bar(Q)_(l))]\left[H_{c}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Sh,Q¯l)] to the cohomology of Rapoport-Zink spaces. Then our knowledge about [Hc(Sh,Q¯l)]HcSh,Q¯l[H_(c)(Sh, bar(Q)_(l))]\left[H_{c}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Sh,Q¯l)] tells us something nontrivial about the cohomology of Rapoport-Zink spaces, and vice versa. This observation turned out to be useful for proving local-global compatibility, i.e., identifying the local Galois action for the Galois representations in Conjecture 4.3 at ramified primes (see [21, cHAP. vII], [65]) and also for understanding the cohomology of basic/nonbasic Rapoport-Zink spaces [2-4,66].
Last but not least, there is Scholze's extension of the Langlands-Kottwitz approach from the hyperspecial level at ppppp to arbitrarily small level subgroup at ppppp. (See Section 6.2 below for another generalization.) One seeks for the following analogue of (3.7), where τ∈τ∈tau in\tau \inτ∈WEpWEpW_(E_(p))W_{E_{\mathfrak{p}}}WEp is a Weil group element with positive valuation, h∈H(G(Qp))h∈HGQph inH(G(Q_(p)))h \in \mathscr{H}\left(G\left(\mathbb{Q}_{p}\right)\right)h∈H(G(Qp)) has support contained in E(Zp)EZpE(Z_(p))\mathcal{E}\left(\mathbb{Z}_{p}\right)E(Zp), and the sum is over the same set of ccccc :
(5.1)tr(f∞,p×h×τ∣[Hc(Sh,Q¯l)])=∑cc(c)Oγ(c)(f∞,p)TOδ(c)(ϕτ,h)(5.1)trâ¡f∞,p×h×τ∣HcSh,Q¯l=∑c c(c)Oγ(c)f∞,pTOδ(c)ϕτ,h{:(5.1)tr(f^(oo,p)xx h xx tau∣[H_(c)(Sh, bar(Q)_(l))])=sum_(c)c(c)O_(gamma(c))(f^(oo,p))TO_(delta(c))(phi_(tau,h)):}\begin{equation*}
\operatorname{tr}\left(f^{\infty, p} \times h \times \tau \mid\left[H_{c}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)\right]\right)=\sum_{\mathrm{c}} c(\mathrm{c}) O_{\gamma(\mathfrak{c})}\left(f^{\infty, p}\right) T O_{\delta(\mathfrak{c})}\left(\phi_{\tau, h}\right) \tag{5.1}
\end{equation*}(5.1)trâ¡(f∞,p×h×τ∣[Hc(Sh,Q¯l)])=∑cc(c)Oγ(c)(f∞,p)TOδ(c)(ϕτ,h)
This has been verified by Scholze [59] for PEL-type data and by Youcis [71] for abeliantype data. As an application, Scholze gave a new proof and characterization of the local Langlands correspondence for GLnGLnGL_(n)\mathrm{GL}_{n}GLn over ppppp-adic fields [60] via a base-change transfer of ϕτ,hϕτ,hphi_(tau,h)\phi_{\tau, h}ϕτ,h. A generalization of the latter to other groups was conjectured in [62] and partially proved for unitary groups by Bertoloni Meli and Youcis [5].
In the proof of (5.1), one can push-forward from arbitrarily small level Kp′Kp′K_(p)^(')K_{p}^{\prime}Kp′ down to hyperspecial level KpKpK_(p)K_{p}Kp at the expense of complicating the coefficient sheaf. Applying the fixed-point formula to this, one can exploit knowledge of the fixed-points (coming from results on the LR conjecture). The main problem is to identify the local terms, which are shown to be encoded by a locally constant compactly supported function ϕτ,hϕτ,hphi_(tau,h)\phi_{\tau, h}ϕτ,h at ppppp constructed from deformation spaces of ppppp-divisible groups with additional structures.
6. SHIMURA VARIETIES WITH BAD REDUCTION, PART II
Let (G,X)(G,X)(G,X)(G, X)(G,X) be a Shimura datum, and ppppp a prime. In this section, we survey generalizations of Sections 2−52−52-52-52−5 in the setup where GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp is allowed to be ramified (thus there may be no unramified Shimura datum of the form (G,X,p,E)(G,X,p,E)(G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) ). We recommend the articles [14,53,56][14,53,56][14,53,56][14,53,56][14,53,56] for introductions to the contents of this section.
6.1. The LR conjecture in the parahoric case
From now until the end of Section 6, assume that (G,X)(G,X)(G,X)(G, X)(G,X) is of abelian type. We fix Kp⊂G(Qp)Kp⊂GQpK_(p)sub G(Q_(p))K_{p} \subset G\left(\mathbb{Q}_{p}\right)Kp⊂G(Qp) a parahoric subgroup, and ppp\mathfrak{p}p a place of EEEEE over ppppp. In this setting, KisinPappas [30] constructed an integral model SKpSKpS_(K_(p))\mathscr{S}_{K_{p}}SKp over OEpOEpO_(E_(p))\mathcal{O}_{E_{\mathfrak{p}}}OEp under a mild hypothesis, which are canonical in the sense of [54].
With the integral model as above, we can state versions of the Langlands-Rapoport conjecture analogous to Conjecture 3.1, cf. [56, §9]. 44^(4){ }^{4}4 One can extend the notion of isogeny classes on SKp(F¯p)SKpF¯pS_(K_(p))( bar(F)_(p))\mathscr{S}_{K_{p}}\left(\overline{\mathbb{F}}_{p}\right)SKp(F¯p) and admissible morphisms ϕ:Q→FGÏ•:Q→FGphi:QrarrF_(G)\phi: \mathbb{Q} \rightarrow \mathbb{F}_{G}Ï•:Q→FG to the parahoric setup, following [30] and [56], respectively. Thus we can consider the set III\mathbb{I}I of isogeny classes and the set JJJ\mathbb{J}J of conjugacy classes of admissible morphisms. The set S(χ)S(χ)S(chi)S(\mathscr{\chi})S(χ) of F¯pF¯pbar(F)_(p)\overline{\mathbb{F}}_{p}F¯p-points in each isogeny class I∈II∈IIinI\mathscr{I} \in \mathbb{I}I∈I is still described by (3.4), with Xp(ℓ)Xp(â„“)X_(p)(â„“)X_{p}(\mathcal{\ell})Xp(â„“) a suitable affine Deligne-Lusztig variety at the parahoric level KpKpK_(p)K_{p}Kp. Analogously we define Sτ(ϕ)SÏ„(Ï•)S_(tau)(phi)S_{\tau}(\phi)SÏ„(Ï•) and Sτ(L)SÏ„(L)S_(tau)(L)S_{\tau}(\mathcal{L})SÏ„(L) for each admissible ϕÏ•phi\phiÏ• and J∈JJ∈JJinJ\mathcal{J} \in \mathbb{J}J∈J, with Xp(ϕ)Xp(Ï•)X_(p)(phi)X_{p}(\phi)Xp(Ï•) in (3.5) also adapted to the parahoric level KpKpK_(p)K_{p}Kp.
Conjecture 6.1. With the above definitions, there exists a G(A∞,p)×ΦZGA∞,p×ΦZG(A^(oo,p))xxPhi^(Z)G\left(\mathbb{A}^{\infty, p}\right) \times \Phi^{\mathbb{Z}}G(A∞,p)×ΦZ-equivariant bijection SKp(F¯p)≅⨆g∈JSτ(J)(J)SKpF¯p≅⨆g∈J SÏ„(J)(J)S_(K_(p))( bar(F)_(p))~=⨆_(ginJ)S_(tau(J))(J)\mathscr{S}_{K_{p}}\left(\overline{\mathbb{F}}_{p}\right) \cong \bigsqcup_{\mathcal{g} \in \mathbb{J}} S_{\tau(\mathcal{J})}(\mathcal{J})SKp(F¯p)≅⨆g∈JSÏ„(J)(J) such that the exact analogue of (LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1), resp. (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) and (LR), holds true.
Van Hoften and Zhou [68,72] proved the following theorem.
Theorem 6.2. Statement (LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1) of Conjecture 6.1 is true if GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp is quasisplit, under a mild technical hypothesis.
The stronger statement (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) is expected to be within reach under the same hypothesis, by extending the argument from [31] to the parahoric setting.
To prove Theorem 6.2, the essential case is when (G,X)(G,X)(G,X)(G, X)(G,X) is of Hodge type. Zhou proves the very special parahoric case of the conjecture. Van Hoften deduces the case of general parahoric subgroup Kp′Kp′K_(p)^(')K_{p}^{\prime}Kp′, contained in a very special parahoric KpKpK_(p)K_{p}Kp, by studying the localization maps from Shimura varieties of level Kp′Kp′K_(p)^(')K_{p}^{\prime}Kp′ and KpKpK_(p)K_{p}Kp, to their respective moduli spaces of local Shtukas. The maps are roughly given by assigning to each abelian variety the associated ppppp-divisible group in terms of the moduli problems. Via the forgetful maps from level Kp′Kp′K_(p)^(')K_{p}^{\prime}Kp′ down to level KpKpK_(p)K_{p}Kp, one can form a commutative square diagram. The central claim is that the diagram is Cartesian, from which the LR conjecture at level Kp′Kp′K_(p)^(')K_{p}^{\prime}Kp′ can be deduced from the known case at level KpKpK_(p)K_{p}Kp. The proof of the claim eventually rests on understanding the irreducible components of Kottwitz-Rapoport strata in the situation of the diagram.
6.2. Semisimple zeta functions and Haines-Kottwitz's test function conjecture
At primes of bad reduction, it is useful to compute the semisimple local zeta factor at ppppp instead of the (true) local factor of the Hasse-Weil zeta function (of Shimura varieties) as the former is more amenable to computation. The latter can be recovered from the former in the cases where the weight-monodromy conjecture is known [55].
4 One can remove the assumption in [56] that Gder Gder G^("der ")G^{\text {der }}Gder is simply connected, by adopting Kisin's formulation in [29] via strictly monoidal categories.
Just like the local zeta factor at ppppp can be computed in terms of the trace of powers of Frobenius on the Frobenius-invariant subspace of the cohomology with compact support, the semisimple local factor can be described in terms of the trace of powers of Frobenius on the derived Frobenius-invariants; such a trace is called the semisimple trace and will be denoted by trsstrsstr^(ss)\operatorname{tr}^{\mathrm{ss}}trss (see [55,$2],[15,$3.1][55,$2],[15,$3.1][55,$2],[15,$3.1][55, \$ 2],[15, \$ 3.1][55,$2],[15,$3.1] ). Thus a key is to establish the following generalization of (3.7), which recovers (3.7) if KpKpK_(p)K_{p}Kp is hyperspecial, due to Haines and Kottwitz [14, 86.1]. The summation is over the same set as in (3.7).
Conjecture 6.3. Let f∞,p∈H(G(A∞,p))f∞,p∈HGA∞,pf^(oo,p)inH(G(A^(oo,p)))f^{\infty, p} \in \mathscr{H}\left(G\left(\mathbb{A}^{\infty, p}\right)\right)f∞,p∈H(G(A∞,p)). For all sufficiently large integers j≫1j≫1j≫1j \gg 1j≫1, there exist test functions ϕHK(j)∈H(G(Qpjr))Ï•HK(j)∈HGQpjrphi_(HK)^((j))inH(G(Q_(p)jr))\phi_{\mathrm{HK}}^{(j)} \in \mathscr{H}\left(G\left(\mathbb{Q}_{p} j r\right)\right)Ï•HK(j)∈H(G(Qpjr)) such that
Moreover, ϕHK(j)Ï•HK(j)phi_(HK)^((j))\phi_{\mathrm{HK}}^{(j)}Ï•HK(j) may be given by an explicit recipe only in terms of local data at ppppp.
When KpKpK_(p)K_{p}Kp is a parahoric subgroup of G(Qp)GQpG(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp), one can be more concrete about ϕHK(j)Ï•HK(j)phi_(HK)^((j))\phi_{\mathrm{HK}}^{(j)}Ï•HK(j) following [14, 87]: ϕHK(j)Ï•HK(j)phi_(HK)^((j))\phi_{\mathrm{HK}}^{(j)}Ï•HK(j) should admit a geometric construction via nearby cycles on local models, as well as a representation-theoretic description in terms of the Langlands correspondence. That the two descriptions for ϕHK(j)Ï•HK(j)phi_(HK)^((j))\phi_{\mathrm{HK}}^{(j)}Ï•HK(j) coincide is the test function conjecture verified by Haines-Richarz [16,17] under a very mild hypothesis. (See [14, §8] for prior and related results.) The proof is based on geometry of mixed-characteristic affine Grassmanians and the geometric Satake equivalence. It remains to combine their theorem with the results in Section 6.1 to obtain new cases of Conjecture 6.3 and its stabilized form, so as to determine the semisimple zeta factor at ppppp. This requires an endoscopic understanding of ϕHK(j)Ï•HK(j)phi_(HK)^((j))\phi_{\mathrm{HK}}^{(j)}Ï•HK(j), cf. [14, §6.2]; a simple exemplary case is demonstrated in [14,$6.3][14,$6.3][14,$6.3][14, \$ 6.3][14,$6.3], where endoscopic problems disappear.
In a related but somewhat different direction (cf. the last two paragraphs in [14, §8.4]), the Langlands-Kottwitz-Scholze approach discussed in Section 5 should extend to the current setup despite the absence of hyperspecial subgroups at ppppp, at least when the results of Section 6.1 are available for some parahoric subgroups.
Problem 6.4. Prove the analogue of (5.1) for general Shimura data ( G,X)G,X)G,X)G, X)G,X) and primes ppppp.
7. IGUSA VARIETIES
Igusa curves were introduced to understand the geometry of modular curves modulo ppppp when the level is divisible by a prime ppppp [24]. The construction has been generalized by Harris-Taylor [21] and Mantovan [44] in the PEL-type case, and most recently to the setup of Kisin-Pappas models for Hodge-type Shimura varieties by Hamacher-Kim [18]. Igusa varieties have a variety of applications to ppppp-adic and mod ppppp modular forms, cohomology of Shimura varieties, the Langlands correspondence, and some more. We refer to the introduction of [36] for further details and references. In this section, we concentrate on computing the ℓâ„“â„“\ellâ„“-adic cohomology of Igusa varieties via an analogue of the LR conjecture.
7.1. The LR conjecture for Igusa varieties
Let (G,X,p,E)(G,X,p,E)(G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) be an unramified Shimura datum of Hodge type, with a fixed embedding of (G,X)(G,X)(G,X)(G, X)(G,X) into a Siegel Shimura datum. Put Kp:=E(Zp)Kp:=EZpK_(p):=E(Z_(p))K_{p}:=\mathscr{E}\left(\mathbb{Z}_{p}\right)Kp:=E(Zp), a hyperspecial subgroup of G(Qp)GQpG(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp). Let AAA\mathscr{A}A denote the abelian scheme over SKpSKpS_(K_(p))\mathscr{S}_{K_{p}}SKp pulled back from the universal abelian scheme over the ambient Siegel moduli scheme. Thus we have a ppppp-divisible group A[p∞]Ap∞A[p^(oo)]\mathcal{A}\left[p^{\infty}\right]A[p∞] over SKpSKpS_(K_(p))\mathscr{S}_{K_{p}}SKp with GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure in some precise sense.
We may and will replace ΣΣSigma\SigmaΣ with an isogenous ppppp-divisible group which is completely slope divisible, since the isomorphism class of IgbIgbIg_(b)\mathrm{Ig}_{b}Igb with the group action is invariant under such a replacement. The advantage of doing so is that IgbIgbIg_(b)\mathrm{Ig}_{b}Igb can be written as the projective limit of finite-type varieties (up to taking perfection, which does not affect cohomology) by trivializing only a finite ppppp-power torsion subgroup and by fixing a level subgroup Kp⊂Kp⊂K^(p)subK^{p} \subsetKp⊂G(A∞,p)GA∞,pG(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p) away from ppppp at a time. With some care, the projective system of varieties can be defined over a common finite field. This enables us to apply the Fujiwara-Varshavsky fixed-point formula to compute the cohomology (with compact support) at each finite level, provided that we understand the structure of Igb(F¯p)Igbâ¡F¯pIg_(b)( bar(F)_(p))\operatorname{Ig}_{b}\left(\overline{\mathbb{F}}_{p}\right)Igbâ¡(F¯p). Thus we are prompted to think about the analogue of the LR conjecture for Igusa varieties.
In analogy with (3.3) and (3.4), keeping the same definition of III\mathbb{I}I and JJJ\mathbb{J}J, we have the partition SIgb(F¯p)=∐ℓ∈ISIgb(ℓ)SIgbF¯p=âˆâ„“∈I SIgb(â„“)S^(Ig_(b))( bar(F)_(p))=âˆ_(â„“inI)S^(Ig_(b))(â„“)S^{\operatorname{Ig}_{b}}\left(\overline{\mathbb{F}}_{p}\right)=\coprod_{\ell \in \mathbb{I}} S^{\mathrm{Ig}_{b}}(\mathcal{\ell})SIgb(F¯p)=âˆâ„“∈ISIgb(â„“) according to isogeny classes of abelian varieties, with
The G(A∞,p)GA∞,pG(A^(oo,p))G\left(\mathbb{A}^{\infty, p}\right)G(A∞,p)-set Xp(d)Xp(d)X^(p)(d)X^{p}(\mathscr{d})Xp(d) is the same as before, but the difference from Section 3 is that XpIg(d)XpIg(d)X_(p)^(Ig)(d)X_{p}^{\mathrm{Ig}}(\mathcal{d})XpIg(d) is no longer an affine Deligne-Lusztig variety but a right Jb(Qp)JbQpJ_(b)(Q_(p))J_{b}\left(\mathbb{Q}_{p}\right)Jb(Qp)-torsor. Turning to the other side of the LRLRLR\mathrm{LR}LR conjecture, let J∈JJ∈JJinJ\mathscr{J} \in \mathbb{J}J∈J. It is natural to impose the so-called bbbbb admissibility condition on JJJ\mathcal{J}J at ppppp, which is the group-theoretic analogue of the condition that a ppppp-divisible group is isogenous to ΣΣSigma\SigmaΣ (with GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp-structure). For bbbbb-admissible LLL\mathcal{L}L, we set
with the same Xp(J)Xp(J)X^(p)(J)X^{p}(\mathcal{J})Xp(J) as in Section 3, and a suitably defined right Jb(Qp)JbQpJ_(b)(Q_(p))J_{b}\left(\mathbb{Q}_{p}\right)Jb(Qp)-torsor XpIgb(L)XpIgb(L)X_(p)^(Ig_(b))(L)X_{p}^{\mathrm{Ig}_{b}}(\mathcal{L})XpIgb(L), where the τÏ„tau\tauÏ„-twisted quotient can be defined again as in Section 3. We are ready to state versions of the LR conjecture for Igusa varieties in parallel with Conjecture 3.1, for unramified Shimura data (G,X,p,E)(G,X,p,E)(G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) of Hodge type.
Conjecture 7.1. There is a bijection of right Jb(Qp)×G(A∞,p)JbQp×GA∞,pJ_(b)(Q_(p))xx G(A^(oo,p))J_{b}\left(\mathbb{Q}_{p}\right) \times G\left(\mathbb{A}^{\infty, p}\right)Jb(Qp)×G(A∞,p)-sets
where {τ(d)}{Ï„(d)}{tau(d)}\{\tau(\mathcal{d})\}{Ï„(d)} over the set of bbbbb-admissible III\mathcal{I}I satisfies the conditions in (LR1)LR1(LR_(1))\left(\mathrm{LR}_{1}\right)(LR1), resp. (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) and (LR).
Mack-Crane proved the following theorem in his thesis [43], where c,c(c)c,c(c)c,c(c)c, c(c)c,c(c), and γ(c)γ(c)gamma(c)\gamma(c)γ(c) are the same as in Section 3, but we impose a bbbbb-admissibility condition on the Kottwitz parameter ccc\mathrm{c}c inherited from the similar condition on LLL\mathcal{L}L, and each ccc\mathrm{c}c gives rise to a conjugacy class of δ′(c)δ′(c)delta^(')(c)\delta^{\prime}(c)δ′(c) in Jb(Qp)JbQpJ_(b)(Q_(p))J_{b}\left(\mathbb{Q}_{p}\right)Jb(Qp), along which we compute the (untwisted) orbital integral Oδ′(c)(ϕp′)Oδ′(c)Ï•p′O_(delta^(')(c))(phi_(p)^('))O_{\delta^{\prime}(c)}\left(\phi_{p}^{\prime}\right)Oδ′(c)(Ï•p′).
Theorem 7.2. The (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0)-version of Conjecture 7.1 is true. Moreover, the following analogue of (3.7) holds true for f∞,p∈H(G(A∞,p))f∞,p∈HGA∞,pf^(oo,p)inH(G(A^(oo,p)))f^{\infty, p} \in \mathscr{H}\left(G\left(\mathbb{A}^{\infty, p}\right)\right)f∞,p∈H(G(A∞,p)) and sufficiently many functions ϕp′∈Ï•p′∈phi_(p)^(')in\phi_{p}^{\prime} \inÏ•p′∈H(Jb(Qp)):HJbQp:H(J_(b)(Q_(p))):\mathscr{H}\left(J_{b}\left(\mathbb{Q}_{p}\right)\right):H(Jb(Qp)):
By "sufficiently many," we mean that the traces for such a set of functions are enough to determine [Hc(Igb,Q¯l)]HcIgb,Q¯l[H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] in the Grothendieck group of G(A∞,p)×Jb(Qp)GA∞,p×JbQpG(A^(oo,p))xxJ_(b)(Q_(p))G\left(\mathbb{A}^{\infty, p}\right) \times J_{b}\left(\mathbb{Q}_{p}\right)G(A∞,p)×Jb(Qp) representations. (The precise condition has to do with twisting by a high power of Frobenius in the Fujiwara-Varshavsky formula.) The proof of the theorem proceeds by carefully adapting the methods of [29,31][29,31][29,31][29,31][29,31] but with significant changes occurring at ppppp, thus often requiring different techniques and arguments.
Formula (7.2) was obtained for some simple PEL-type Shimura varieties in [21, cHAP. 5] and [63] without formulating and proving the LR conjecture. In contrast, the above theorem represents the first LR-style approach to Igusa varieties, giving it two advantages. Firstly, the new approach makes the similarities between Shimura and Igusa varieties transparent. An important consequence is that the hard-won statement (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) for Shimura varieties can be transferred to the Igusa side. (If we had the full (LR) for Shimura varieties, then that would carry over to Igusa varieties, too.) Going from (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) for Igusa varieties to (7.2) is mostly the same as for Shimura varieties. Secondly, just like for Shimura varieties, the LR-style approach makes it feasible to extend the theorem to the abelian-type case, cf. Remark 3.4. (This extension has not been worked out, yet.) It should also be possible to go beyond good reduction and work in the setup of Kisin-Pappas models (Section 6). For the sake of proposing a problem, we can be even more general but still stick to (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0) rather than (LR) as this should suffice for most applications:
Problem 7.3. Construct Igusa varieties modulo ppppp for all Shimura data (G,X)(G,X)(G,X)(G, X)(G,X) and all primes ppppp. Prove the (LR0)LR0(LR_(0))\left(\mathrm{LR}_{0}\right)(LR0)-version of Conjecture 7.1, thereby deduce formula (7.2).
Assuming a positive answer (known in the setting of Theorem 7.2), the next step is to unconditionally stabilize (7.2) into the following form:
The formula is an exact analogue of (3.8). Indeed, fe,∞,pfe,∞,pf^(e,oo,p)f^{e, \infty, p}fe,∞,p and f∞ef∞ef_(oo)^(e)f_{\infty}^{e}f∞e are constructed in the same way. However, fpe,fpe,f_(p)^(e,)f_{p}^{e,}fpe, is constructed from ϕp′Ï•p′phi_(p)^(')\phi_{p}^{\prime}Ï•p′ via a "nonstandard" transfer of functions this is the main novelty in the stabilization. Even when GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp is a product of general linear groups so that local endoscopy at ppppp disappears, the transfer goes to GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp from an inner form of a Levi subgroup of GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp. The transfer was constructed and studied in [64] in a somewhat ad hoc manner, and later streamlined in [2]. Unfortunately, both papers make a set of technical hypotheses, to be removed in the work in progress with Bertoloni Meli.
7.2. Applications
The stabilization (7.3) is a significant step towards the following:
Problem 7.4. Obtain a decomposition of [Hc(Igb,Q¯l)]HcIgb,Q¯l[H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] according to automorphic representations of GGGGG and its endoscopic groups, and describe each piece in the decomposition.
Just from the definition of Igusa varieties, it is not even clear whether the entirety of [Hc(Igb,Q¯l)]HcIgb,Q¯l[H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] can be understood through automorphic representations. For Shimura varieties over CCC\mathbb{C}C, the connection is made through Matsushima's formula and its generalizations, but there is no analogue for Igusa varieties.
We have a concrete answer for some simple PEL-type Shimura varieties arising from (G,X)(G,X)(G,X)(G, X)(G,X) such that (i) endoscopy for GGGGG disappears over QQQ\mathbb{Q}Q and QpQpQ_(p)\mathbb{Q}_{p}Qp, (ii) GGGGG is anisotropic modulo center over QQQ\mathbb{Q}Q, and (iii) GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp is a product of general linear groups. Recall that JbJbJ_(b)J_{b}Jb is an inner form of a QpQpQ_(p)\mathbb{Q}_{p}Qp-rational Levi subgroup, say MbMbM_(b)M_{b}Mb, of GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp. In fact, bbbbb determines a particular parabolic subgroup PbopPbopP_(b)^(op)P_{b}^{\mathrm{op}}Pbop containing MbMbM_(b)M_{b}Mb as a Levi component. Write Red bb^(b){ }^{b}b for the composite morphism on the Grothendieck group of representations
where the first map is the Jacquet module relative to Pbop Pbop P_(b)^("op ")P_{b}^{\text {op }}Pbop (up to a character twist), and the second is Badulescu's Langlands-Jacquet map. The answer to Problem 7.4 in this setting is given by [21, тнм. V.5.4] and [66, тнм. 6.7]:
(7.5)[Hc(Igb,Q¯l)]=[RedbHc(Sh,Q¯l)] in Groth(G(A∞,p)×Jb(Qp))(7.5)HcIgb,Q¯l=Redbâ¡HcSh,Q¯l in Grothâ¡GA∞,p×JbQp{:(7.5)[H_(c)(Ig_(b), bar(Q)_(l))]=[Red^(b)H_(c)(Sh, bar(Q)_(l))]quad" in "Groth(G(A^(oo,p))xxJ_(b)(Q_(p))):}\begin{equation*}
\left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right]=\left[\operatorname{Red}^{b} H_{c}\left(\operatorname{Sh}, \overline{\mathbb{Q}}_{l}\right)\right] \quad \text { in } \operatorname{Groth}\left(G\left(\mathbb{A}^{\infty, p}\right) \times J_{b}\left(\mathbb{Q}_{p}\right)\right) \tag{7.5}
\end{equation*}(7.5)[Hc(Igb,Q¯l)]=[Redbâ¡Hc(Sh,Q¯l)] in Grothâ¡(G(A∞,p)×Jb(Qp))
Since Hc(Sh,Q¯l)HcSh,Q¯lH_(c)(Sh, bar(Q)_(l))H_{c}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)Hc(Sh,Q¯l) is well understood by Matsushima's formula via relative Lie algebra cohomology, (7.5) is indeed a satisfactory answer for [Hc(Igb,Q¯l)]HcIgb,Q¯l[H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)]. Through Mantovan's formula, (7.5) sheds light on the cohomology of Rapoport-Zink spaces [3,66], cf. Section 5.
When GQpGQpG_(Q_(p))G_{\mathbb{Q}_{p}}GQp is still a product of general linear groups but GGGGG exhibits endoscopy over QQQ\mathbb{Q}Q, the formula for [Hc(Igb,Q¯l)]HcIgb,Q¯l[H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] is no longer as simple as (7.5). Computing the formula in certain endoscopic cases was crucial in the proof of local-global compatibility in [65], cf. Section 5. (See [67,$6][67,$6][67,$6][67, \$ 6][67,$6] for an expository account.)
In general, Problem 7.4 seems out of reach. Firstly, just like for Shimura varieties, the lack of endoscopic classification is a major obstacle. Secondly, a new difficulty in the Igusa setup is that the analogue of Problem 4.1 has no conceptual reason to have a positive answer (cf. last paragraph of Section 4.1), and the analogue of Problem 4.2 is even less clear. (The second point is related to the question at the end of this section.) Assuming that both issues
go away, a conjectural formula for [Hc(Igb,Q¯l)]HcIgb,Q¯l[H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\mathrm{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] has been given in [2] under some hypotheses on GGGGG, resembling Kottwitz's conjectural formula for [IH(Sh¯,Q¯l)]IHâ¡Sh¯,Q¯l[IH( bar(Sh), bar(Q)_(l))]\left[\operatorname{IH}\left(\overline{\mathrm{Sh}}, \overline{\mathbb{Q}}_{l}\right)\right][IHâ¡(Sh¯,Q¯l)] in [32, §10]. The formula for [Hc(Igb,Q¯l)]HcIgb,Q¯l[H_(c)(Ig_(b), bar(Q)_(l))]\left[H_{c}\left(\mathrm{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)\right][Hc(Igb,Q¯l)] is far more complicated than (7.5) and involves endoscopic versions of RedbRedbRed^(b)\operatorname{Red}^{b}Redb. (In the stable case, which is the simplest, the correct analogue of (7.4) is the Jacquet module followed by a stable transfer between inner forms.) A main observation in [2] is that the endoscopic versions of RedbRedbRed^(b)\operatorname{Red}^{b}Redb should interact with the cohomology of Rapoport-Zink spaces (and their generalizations) in an interesting way by a global reason.
For an unconditional result towards Problem 7.4, we managed to compute the G(A∞,p)×Jb(Qp)GA∞,p×JbQpG(A^(oo,p))xxJ_(b)(Q_(p))G\left(\mathbb{A}^{\infty, p}\right) \times J_{b}\left(\mathbb{Q}_{p}\right)G(A∞,p)×Jb(Qp)-module H0(Igb,Q¯l)H0Igb,Q¯lH^(0)(Ig_(b), bar(Q)_(l))H^{0}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)H0(Igb,Q¯l) for unramified Shimura data (G,X,p,E)(G,X,p,E)(G,X,p,E)(G, X, p, \mathcal{E})(G,X,p,E) of Hodge type in joint work with Kret [36], when bbbbb is nonbasic 55^(5){ }^{5}5 (in every QQQ\mathbb{Q}Q-simple factor of the adjoint group of GGGGG ). In analogy with (7.5), the result may be expressed as
Here H0(Sh,Q¯l)H0Sh,Q¯lH^(0)(Sh, bar(Q)_(l))H^{0}\left(\mathrm{Sh}, \overline{\mathbb{Q}}_{l}\right)H0(Sh,Q¯l) has a well-known description in terms of 1-dimensional automorphic representations of G(A)G(A)G(A)G(\mathbb{A})G(A), and RedbRedbRed^(b)\operatorname{Red}^{b}Redb is unequivocally defined for 1-dimensional representations of G(Qp)GQpG(Q_(p))G\left(\mathbb{Q}_{p}\right)G(Qp) (which are always stable). The proof starts from the results of Section 7.1. The main point is to get around the two essential difficulties mentioned in the last paragraph, by incorporating asymptotic and inductive arguments to extract the H0H0H^(0)H^{0}H0-part from a very complicated identity coming from (stabilized) trace formulas. The study of H0H0H^(0)H^{0}H0 was motivated by geometric applications to the irreducibility of Igusa varieties and to the discrete Hecke orbit conjecture. The reader is referred to [36] for details and further references. Similar geometric results were independently obtained by van Hoften and Xiao [68,69[68,69[68,69[68,69[68,69 ] via a more geometric approach without using automorphic forms or trace formulas.
We conclude this section with an unrefined question. Igusa varieties (at finite level) are almost never proper varieties, so the answer to Problem 7.4 does not determine Hci(Igb,Q¯l)HciIgb,Q¯lH_(c)^(i)(Ig_(b), bar(Q)_(l))H_{c}^{i}\left(\operatorname{Ig}_{b}, \overline{\mathbb{Q}}_{l}\right)Hci(Igb,Q¯l) for i≥0i≥0i >= 0i \geq 0i≥0 due to possible cancelations in the Grothendieck group. (For improper Shimura varieties, the intersection cohomology is free from such a cancelation thanks to purity.) Thus we can ask whether there are useful compactifications of Igusa varieties to help us understand the cohomology more precisely. This was undertaken by Mantovan [45] in a special case (with a different goal). In general, it is unclear how to proceed even when Shimura varieties are proper. If a strategy is found in that case, it may be possible to deal with improperness of Shimura varieties by virture of Caraiani-Scholze's partial compactification [8][8][8][8][8].
ACKNOWLEDGMENTS
Special thanks are due to Michael Harris, Robert Kottwitz, and Richard Taylor for introducing the author to Shimura varieties and the Langlands program, and patiently answering his questions over many years. The author is grateful to Mark Kisin, Arno Kret, and Yihang Zhu for sharing their insight through years of recent and ongoing collaboration
5 If bbbbb is basic then Igusa varieties are 0 -dimensional and the picture is quite different from (7.6), cf. [36, §1.6]
that this paper is largely based on. He is also thankful for exciting mathematical journeys to the other coworkers: Sara Arias-de Reyna, Ana Caraiani, Luis Dieulefait, Matthew Emerton, Jessica Fintzen, Toby Gee, David Geraghty, Wushi Goldring, Julia Gordon, Junehyuk Jung, Tasho Kaletha, Julee Kim, Keerthi Madapusi Pera, Simon Marshall, Alberto Minguez, Vytautas Paškūnas, Peter Sarnak, Peter Scholze, Nicolas Templier, PaulJames White, and Gabor Wiese. Finally, he thanks Alex Youcis for comments to improve the exposition of the paper.
FUNDING
This work was partially supported by NSF grant DMS-1802039/2101688, NSF RTG grant DMS-1646385, and a Miller Professorship.
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SUG WOO SHIN
Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA, and Korea Institute for Advanced Study, Seoul 02455, Republic of Korea, sug.woo.shin@berkeley.edu
THE CONGRUENT NUMBER PROBLEM AND ELLIPTIC CURVES
YE TIAN
ABSTRACT
The Birch and Swinnerton-Dyer (BSD) conjecture and Goldfeld conjecture are fundamental problems in the arithmetic of elliptic curves. The congruent number problem (CNP) is one of the oldest problems in number theory which is, for each integer nnnnn, to find all the rational right triangles of area nnnnn. It is equivalent to finding all rational points on the elliptic curve E(n):ny2=x3−xE(n):ny2=x3−xE^((n)):ny^(2)=x^(3)-xE^{(n)}: n y^{2}=x^{3}-xE(n):ny2=x3−x. The BSD conjecture for E(n)E(n)E^((n))E^{(n)}E(n) solves CNP, and Goldfeld conjecture for this elliptic curve family solves CNP for integers with probability one. In this article, we introduce some recent progress on these conjectures and problems.
A positive rational number nnnnn is called a congruent number if the following equivalent conditions hold:
(i) There exists a rational number xxxxx such that x2±nx2±nx^(2)+-nx^{2} \pm nx2±n are squares of rational numbers.
(ii) There exists a right triangle with rational side lengths (called a rational right triangle) whose area is nnnnn.
In his book Liber Quadratorum published in 1225, Fibonacci (1175-1250) named an integer satisfying (i) a "congruum" from the Latin, which means to meet together, since the three squares x2−n,x2x2−n,x2x^(2)-n,x^(2)x^{2}-n, x^{2}x2−n,x2, and x2+nx2+nx^(2)+nx^{2}+nx2+n are congruent modulo nnnnn.
The congruent number problem (CNP, for short) is to determine, in finitely many steps, whether or not a given rational number is a congruent number, and, if it is, find all the corresponding xxxxx in (i) or rational right triangles in (ii). No such algorithm has ever been found. The Persian mathematician Al-Karaji (953-1029), perhaps the first mathematician, stated this problem in terms of (i). A similar question appeared in his Arabic translation of the work of Diophantus in Greek. In an Arab manuscript of the tenth century, Mohammed Ben Alhocain realized the equivalence between (i) and (ii) and stated that this problem is "the principal object of the theory of rational right triangles" (see Dickson's book [20, chAP.
XVI, P. 459]).
Recall that any rational Pythagorean triple has the following form:
2abt,(a2−b2)t,(a2+b2)t2abt,a2−b2t,a2+b2t2abt,quad(a^(2)-b^(2))t,quad(a^(2)+b^(2))t2 a b t, \quad\left(a^{2}-b^{2}\right) t, \quad\left(a^{2}+b^{2}\right) t2abt,(a2−b2)t,(a2+b2)t
for a unique (a,b,t)(a,b,t)(a,b,t)(a, b, t)(a,b,t), where ttttt is a positive rational number and a>ba>ba > ba>ba>b are two coprime positive integers with 2∤(a+b)2∤(a+b)2∤(a+b)2 \nmid(a+b)2∤(a+b). We call a rational Pythagorean triple primitive if t=1t=1t=1t=1t=1, i.e., its triangle has coprime integral side lengths. It follows that nnnnn is a congruent number if and only if nnnnn has the same square-free part as ab(a+b)(a−b)ab(a+b)(a−b)ab(a+b)(a-b)a b(a+b)(a-b)ab(a+b)(a−b), for some integers aaaaa and bbbbb. For example, by taking (a,b)=(5,4),(2,1)(a,b)=(5,4),(2,1)(a,b)=(5,4),(2,1)(a, b)=(5,4),(2,1)(a,b)=(5,4),(2,1), and (16,9)(16,9)(16,9)(16,9)(16,9), note that 5,6,75,6,75,6,75,6,75,6,7 are congruent numbers with corresponding triangles (20/3,3/2,41/6),(3,4,5)(20/3,3/2,41/6),(3,4,5)(20//3,3//2,41//6),(3,4,5)(20 / 3,3 / 2,41 / 6),(3,4,5)(20/3,3/2,41/6),(3,4,5), and (24/5,35/12,337/60)(24/5,35/12,337/60)(24//5,35//12,337//60)(24 / 5,35 / 12,337 / 60)(24/5,35/12,337/60). To consider CNP, it is enough to consider square-free integers. In Liber Quadratorum, Fibonacci constructed these right triangles and also claimed that 1 is not a congruent number, but did not give a proof.
In 1640, Fermat discovered his infinite descent method to show that 1,2,31,2,31,2,31,2,31,2,3 are noncongruent numbers. The same method could be employed to find more noncongruent numbers, for example, any prime p≡3(mod8)p≡3(mod8)p-=3(mod8)p \equiv 3(\bmod 8)p≡3(mod8). In fact, suppose such a prime ppppp is a congruent number, then there exists a primitive Pythagorean triple (a2−b2,2ab,a2+b2)a2−b2,2ab,a2+b2(a^(2)-b^(2),2ab,a^(2)+b^(2))\left(a^{2}-b^{2}, 2 a b, a^{2}+b^{2}\right)(a2−b2,2ab,a2+b2) whose area ab(a+b)(a−b)ab(a+b)(a−b)ab(a+b)(a-b)a b(a+b)(a-b)ab(a+b)(a−b) has the square-free part ppppp. Assume the area is minimal. Since a,b,a+b,a−ba,b,a+b,a−ba,b,a+b,a-ba, b, a+b, a-ba,b,a+b,a−b are coprime to each other, by modulo 8 consideration, we have
for some positive integers r,s,u,vr,s,u,vr,s,u,vr, s, u, vr,s,u,v. Note that the Pythagorean triple (u−v,u+v,2r)(u−v,u+v,2r)(u-v,u+v,2r)(u-v, u+v, 2 r)(u−v,u+v,2r) is with smaller area, a contradiction.
More examples of congruent and noncongruent numbers (gray for non-congruent numbers) were found:
From the table, one may conjecture that all the positive integers congruent to 5,6,75,6,75,6,75,6,75,6,7 modulo 8 are congruent numbers (conjectured by Alter, Curtz, and Kubota in [2]) and the density of positive integers congruent to 1,2,31,2,31,2,31,2,31,2,3 modulo 8 being non-congruent is one.
The arithmetic of elliptic curves, in particular the BSD conjecture and the Goldfeld conjecture, provides a systematical and deeper point of view to study CNP. We now recall these conjectures to introduce notation.
For an elliptic curve AAAAA over a number field FFFFF, the set A(F)A(F)A(F)A(F)A(F) of rational points has a finitely generated abelian group structure by Mordell-Weil theorem. Its rank is denoted by rankZA(F)rankZâ¡A(F)rank_(Z)A(F)\operatorname{rank}_{\mathbb{Z}} A(F)rankZâ¡A(F). The Hasse-Weil L-function L(s,A/F)L(s,A/F)L(s,A//F)L(s, A / F)L(s,A/F) of AAAAA is defined as an Euler product and conjectured to be entire and to satisfy a functional equation. The vanishing order of L(s,A/F)L(s,A/F)L(s,A//F)L(s, A / F)L(s,A/F) at s=1s=1s=1s=1s=1 denoted by ords=1L(s,A/F)ords=1â¡L(s,A/F)ord_(s=1)L(s,A//F)\operatorname{ord}_{s=1} L(s, A / F)ords=1â¡L(s,A/F) is called the analytic rank of A/FA/FA//FA / FA/F. When F=QF=QF=QF=\mathbb{Q}F=Q, the conjecture is known by the work of Wiles [54], et al., and the functional equation is given by
Λ(s,A/Q):=NAs/2⋅2(2π)−sΓ(s)L(s,A/Q)=ϵ(A)Λ(2−s,A/Q)Λ(s,A/Q):=NAs/2â‹…2(2Ï€)−sΓ(s)L(s,A/Q)=ϵ(A)Λ(2−s,A/Q)Lambda(s,A//Q):=N_(A)^(s//2)*2(2pi)^(-s)Gamma(s)L(s,A//Q)=epsilon(A)Lambda(2-s,A//Q)\Lambda(s, A / \mathbb{Q}):=N_{A}^{s / 2} \cdot 2(2 \pi)^{-s} \Gamma(s) L(s, A / \mathbb{Q})=\epsilon(A) \Lambda(2-s, A / \mathbb{Q})Λ(s,A/Q):=NAs/2â‹…2(2Ï€)−sΓ(s)L(s,A/Q)=ϵ(A)Λ(2−s,A/Q)
where NA∈Z≥1NA∈Z≥1N_(A)inZ_( >= 1)N_{A} \in \mathbb{Z}_{\geq 1}NA∈Z≥1 is the conductor of A/QA/QA//QA / \mathbb{Q}A/Q, and ϵ(A)∈{±1}ϵ(A)∈{±1}epsilon(A)in{+-1}\epsilon(A) \in\{ \pm 1\}ϵ(A)∈{±1} is the root number.
Conjecture 1 (BSD). Let A be an elliptic curve over a number field FFFFF. Then the following holds:
(1) rankZA(F)=ords=1L(s,A/F)rankZâ¡A(F)=ords=1â¡L(s,A/F)rank_(Z)A(F)=ord_(s=1)L(s,A//F)\operatorname{rank}_{\mathbb{Z}} A(F)=\operatorname{ord}_{s=1} L(s, A / F)rankZâ¡A(F)=ords=1â¡L(s,A/F).
(2) The Tate-Shafarevich group ⨿(A/F)⨿(A/F)⨿(A//F)\amalg(A / F)⨿(A/F) is finite. For r=ords=1L(s,A/F)r=ords=1â¡L(s,A/F)r=ord_(s=1)L(s,A//F)r=\operatorname{ord}_{s=1} L(s, A / F)r=ords=1â¡L(s,A/F),
For a prime ppppp, we call the equality of ppppp-valuation on both sides the ppppp-part BSD formula.
One significant fact related to the BSD conjecture for an elliptic curve AAAAA over QQQ\mathbb{Q}Q is that if it holds, then there will be an effective algorithm to compute generators of A(Q)A(Q)A(Q)A(\mathbb{Q})A(Q) [39]. It is easy to see that a positive integer nnnnn is a congruent number if and only if the elliptic curve (called a congruent elliptic curve)
E(n):ny2=x3−xE(n):ny2=x3−xE^((n)):ny^(2)=x^(3)-xE^{(n)}: n y^{2}=x^{3}-xE(n):ny2=x3−x
has Mordell-Weil group E(n)(Q)E(n)(Q)E^((n))(Q)E^{(n)}(\mathbb{Q})E(n)(Q) of positive rank. There exists a one-to-one correspondence between rational right triangles with area nnnnn and nontorsion rational points of E(n)E(n)E^((n))E^{(n)}E(n). In particular, the BSD conjecture for E(n)E(n)E^((n))E^{(n)}E(n) would solve the CNP.
A fundamental result on the BSD conjecture was obtained by Coates-Wiles [18], Rubin [43], Gross-Zagier [24], and Kolyvagin [36]: If ord s=1L(s,A/Q)≤1s=1L(s,A/Q)≤1_(s=1)L(s,A//Q) <= 1{ }_{s=1} L(s, A / \mathbb{Q}) \leq 1s=1L(s,A/Q)≤1, then
rankZA(Q)=ords=1L(s,A/Q)rankZâ¡A(Q)=ords=1â¡L(s,A/Q)rank_(Z)A(Q)=ord_(s=1)L(s,A//Q)\operatorname{rank}_{\mathbb{Z}} A(\mathbb{Q})=\operatorname{ord}_{s=1} L(s, A / \mathbb{Q})rankZâ¡A(Q)=ords=1â¡L(s,A/Q)
and #⨿(A/Q)<∞#⨿(A/Q)<∞#⨿(A//Q) < oo\# \amalg(A / \mathbb{Q})<\infty#⨿(A/Q)<∞. There are several results on the ppppp-part BSD formula, including Rubin [43], Kato [33], Kolyvagin [36], Skinner-Urban [46], Zhang [56], Jetchev-Skinner-Wan [31]. The full BSD conjecture was verified for a subfamily of congruent elliptic curves, which have both algebraic and analytic rank one.
Theorem 2([37])2([37])2([37])2([37])2([37]). Let n≡5(mod8)n≡5(mod8)n-=5(mod8)n \equiv 5(\bmod 8)n≡5(mod8) be a square-free positive integer, all of whose prime factors are congruent to 1 modulo 4 . Assume that Q(−n)Q(−n)Q(sqrt(-n))\mathbb{Q}(\sqrt{-n})Q(−n) has no ideal class of order 4 , then E(n):y2=x3−n2xE(n):y2=x3−n2xE^((n)):y^(2)=x^(3)-n^(2)xE^{(n)}: y^{2}=x^{3}-n^{2} xE(n):y2=x3−n2x has both algebraic and analytic rank 1 and the full BSD conjecture holds.
For the above congruent number elliptic curves, the 2-part of the BSD formula is proved in [51], [50]. The ppppp-part of the BSD formula, when p≥3,p∤np≥3,p∤np >= 3,p∤np \geq 3, p \nmid np≥3,p∤n, is the consequence of works by Perrin-Riou [42], Kobayashi [35], etal. The ppppp-part of the BSD formula, when p≡1(mod4),p∣np≡1(mod4),p∣np-=1(mod4),p∣np \equiv 1(\bmod 4), p \mid np≡1(mod4),p∣n, is proved in Li-Liu-Tian [37]. The generalization of Kobayashi's work to potential supersingular primes together with the argument of Perrin-Riou [42], also implies the ppppp-part BSD formula for primes ppppp of potential supersingular reduction (see [41]).
There is a conjecture on statistical behaviors of analytic ranks for a quadratic twist family of elliptic curves. For an elliptic curve y2=f(x)y2=f(x)y^(2)=f(x)y^{2}=f(x)y2=f(x) over FFFFF, its quadratic twist family consists of elliptic curves ny2=f(x)ny2=f(x)ny^(2)=f(x)n y^{2}=f(x)ny2=f(x) with n∈F×n∈F×n inF^(xx)n \in F^{\times}n∈F×. Based on minimalist principle, Goldfeld proposed the following:
Conjecture 3 (Goldfeld [14,23]). Let ε∈{±1}ε∈{±1}epsi in{+-1}\varepsilon \in\{ \pm 1\}ε∈{±1} and AAA\mathscr{A}A be a quadratic twist family of elliptic curves over FFFFF. Then, ordered by norms of conductors, among the quadratic twists A∈AA∈AA inAA \in \mathcal{A}A∈A with ϵ(A)=εϵ(A)=εepsilon(A)=epsi\epsilon(A)=\varepsilonϵ(A)=ε,
Prob(ords=1L(s,A/F)=0)(resp.Prob(ords=1L(s,A/F)=1))Probâ¡ords=1â¡L(s,A/F)=0resp.Probâ¡ords=1â¡L(s,A/F)=1Prob(ord_(s=1)L(s,A//F)=0)quad(resp.Prob(ord_(s=1)L(s,A//F)=1))\operatorname{Prob}\left(\operatorname{ord}_{s=1} L(s, A / F)=0\right) \quad\left(\operatorname{resp} . \operatorname{Prob}\left(\operatorname{ord}_{s=1} L(s, A / F)=1\right)\right)Probâ¡(ords=1â¡L(s,A/F)=0)(resp.Probâ¡(ords=1â¡L(s,A/F)=1))
is one if ε=+1ε=+1epsi=+1\varepsilon=+1ε=+1 (resp. -1 ). In particular, if F=QF=QF=QF=\mathbb{Q}F=Q, as A runs over a quadratic twist family of elliptic curves,
Prob(ords=1L(s,A/Q)=r)Probâ¡ords=1â¡L(s,A/Q)=rProb(ord_(s=1)L(s,A//Q)=r)\operatorname{Prob}\left(\operatorname{ord}_{s=1} L(s, A / \mathbb{Q})=r\right)Probâ¡(ords=1â¡L(s,A/Q)=r)
is equal to 1/21/21//21 / 21/2 for r=0,1r=0,1r=0,1r=0,1r=0,1, and 0 for r≥2r≥2r >= 2r \geq 2r≥2.
We refer to ϵ=1ϵ=1epsilon=1\epsilon=1ϵ=1 (resp. -1 ) case of the conjecture as the even (resp. odd) parity Goldfeld conjecture. The significance of Goldfeld conjecture is that, together with the GrossZagier formula (see Section 2), it solves the problem of finding generators of A(Q)A(Q)A(Q)A(\mathbb{Q})A(Q) for density-one elliptic curves AAAAA in a quadratic twist family.
Conjecture 4 (Goldfeld [23], Katz-Sarnak [34], etc.). Let A run over all elliptic curves over a fixed number field FFFFF as ordered by height, then
Prob(ords=1L(s,A/F)=r)Probâ¡ords=1â¡L(s,A/F)=rProb(ord_(s=1)L(s,A//F)=r)\operatorname{Prob}\left(\operatorname{ord}_{s=1} L(s, A / F)=r\right)Probâ¡(ords=1â¡L(s,A/F)=r)
is equal to 1/21/21//21 / 21/2 for r=0,1r=0,1r=0,1r=0,1r=0,1, and 0 for r≥2r≥2r >= 2r \geq 2r≥2.
For n∈Z≥0n∈Z≥0n inZ_( >= 0)n \in \mathbb{Z}_{\geq 0}n∈Z≥0, we have
ϵ(E(n))={+1,n≡1,2,3(mod8)−1,n≡5,6,7(mod8)ϵE(n)=+1,n≡1,2,3(mod8)−1,n≡5,6,7(mod8)epsilon(E^((n)))={[+1",",n-=1","2","3(mod8)],[-1",",n-=5","6","7(mod8)]:}\epsilon\left(E^{(n)}\right)= \begin{cases}+1, & n \equiv 1,2,3(\bmod 8) \\ -1, & n \equiv 5,6,7(\bmod 8)\end{cases}ϵ(E(n))={+1,n≡1,2,3(mod8)−1,n≡5,6,7(mod8)
The central LLL\mathrm{L}L-value of E(n)E(n)E^((n))E^{(n)}E(n) is related to the following ternary quadratic equation by Tunnell [52]: For a positive square-free integer nnnnn, let a=1a=1a=1a=1a=1 if nnnnn is odd and a=2a=2a=2a=2a=2 if nnnnn is even. Consider the equation
2ax2+y2+8z2=n/a,x,y,z∈Z2ax2+y2+8z2=n/a,x,y,z∈Z2ax^(2)+y^(2)+8z^(2)=n//a,quad x,y,z inZ2 a x^{2}+y^{2}+8 z^{2}=n / a, \quad x, y, z \in \mathbb{Z}2ax2+y2+8z2=n/a,x,y,z∈Z
Let Σ(n)Σ(n)Sigma(n)\Sigma(n)Σ(n) be the set of its solutions and let
L(n)=#{(x,y,z)∈Σ(n)|2|z}−#{(x,y,z)∈Σ(n)∣2∤z}L(n)=#{(x,y,z)∈Σ(n)|2|z}−#{(x,y,z)∈Σ(n)∣2∤z}L(n)=#{(x,y,z)in Sigma(n)|2|z}-#{(x,y,z)in Sigma(n)∣2∤z}\mathscr{L}(n)=\#\{(x, y, z) \in \Sigma(n)|2| z\}-\#\{(x, y, z) \in \Sigma(n) \mid 2 \nmid z\}L(n)=#{(x,y,z)∈Σ(n)|2|z}−#{(x,y,z)∈Σ(n)∣2∤z}
It is easy to see that L(n)=0L(n)=0L(n)=0\mathscr{L}(n)=0L(n)=0 for positive n≡5,6,7(mod8)n≡5,6,7(mod8)n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8). Tunnell proved that for nnnnn positive square-free, L(n)≠0L(n)≠0L(n)!=0\mathscr{L}(n) \neq 0L(n)≠0 if and only if L(1,E(n))≠0L1,E(n)≠0L(1,E^((n)))!=0L\left(1, E^{(n)}\right) \neq 0L(1,E(n))≠0. The BSD conjecture predicts the following:
Conjecture A. A positive square-free integer nnnnn is a congruent number if and only if L(n)=0L(n)=0L(n)=0\mathscr{L}(n)=0L(n)=0. In particular, any positive integer n≡5,6,7(mod8)n≡5,6,7(mod8)n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8) is a congruent number.
One can determine whether L(n)=0L(n)=0L(n)=0\mathscr{L}(n)=0L(n)=0 in finitely many steps, yet there is no algorithm to find all the rational points of E(n)E(n)E^((n))E^{(n)}E(n). Tunnell's work was recently generalized to any given quadratic twist family of elliptic curves over QQQ\mathbb{Q}Q in [26].
The even Goldfeld conjecture for the family E(n)E(n)E^((n))E^{(n)}E(n) can be stated as follows:
Conjecture B1. Among all square-free positive integers n≡1,2,3(mod8)n≡1,2,3(mod8)n-=1,2,3(mod8)n \equiv 1,2,3(\bmod 8)n≡1,2,3(mod8), the subset of nnnnn with L(n)≠0L(n)≠0L(n)!=0\mathscr{L}(n) \neq 0L(n)≠0 has density one.
For an elliptic curve A/QA/QA//QA / \mathbb{Q}A/Q with root number -1 , the BSD conjecture predicts that A(Q)A(Q)A(Q)A(\mathbb{Q})A(Q) has an infinite-order point. Heegner point construction provides a systematic method to construct rational points. We now give a concrete construction for congruent elliptic curves E(n)E(n)E^((n))E^{(n)}E(n) with n≡5,6,7(mod8)n≡5,6,7(mod8)n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8). Denote by EEEEE the elliptic curve y2=x3−xy2=x3−xy^(2)=x^(3)-xy^{2}=x^{3}-xy2=x3−x that has conductor 32. The Abel-Jacobi map induces the complex uniformization
E(C)≃C/ΛE,z↦∫Ozdx/2yE(C)≃C/ΛE,z↦∫Oz dx/2yE(C)≃C//Lambda_(E),quad z|->int_(O)^(z)dx//2yE(\mathbb{C}) \simeq \mathbb{C} / \Lambda_{E}, \quad z \mapsto \int_{O}^{z} d x / 2 yE(C)≃C/ΛE,z↦∫Ozdx/2y
where ΛE={∫γdx/2y∣γ∈H1(E(C),Z)}⊂CΛE=∫γ dx/2y∣γ∈H1(E(C),Z)⊂CLambda_(E)={int_(gamma)dx//2y∣gamma inH_(1)(E(C),Z)}subC\Lambda_{E}=\left\{\int_{\gamma} d x / 2 y \mid \gamma \in H_{1}(E(\mathbb{C}), \mathbb{Z})\right\} \subset \mathbb{C}ΛE={∫γdx/2y∣γ∈H1(E(C),Z)}⊂C is the period lattice. Denote by ϕÏ•phi\phiÏ• the newform of weight 2 and level Γ0(32)Γ0(32)Gamma_(0)(32)\Gamma_{0}(32)Γ0(32) associated to EEEEE. Let fffff be the analytic map
induced by the above complex uniformization E(C)≃C/ΛEE(C)≃C/ΛEE(C)≃C//Lambda_(E)E(\mathbb{C}) \simeq \mathbb{C} / \Lambda_{E}E(C)≃C/ΛE and
H→C:τ↦∫i∞τ2πiϕ(z)dzH→C:τ↦∫i∞τ 2Ï€iÏ•(z)dzHrarrC:tau|->int_(i oo)^(tau)2pi i phi(z)dz\mathscr{H} \rightarrow \mathbb{C}: \tau \mapsto \int_{i \infty}^{\tau} 2 \pi i \phi(z) d zH→C:τ↦∫i∞τ2Ï€iÏ•(z)dz
We now give a construction of Heegner points. For nnnnn a positive square-free integer ≡5,6,7(mod8)≡5,6,7(mod8)-=5,6,7(mod8)\equiv 5,6,7(\bmod 8)≡5,6,7(mod8), let K=Q(−n)K=Q(−n)K=Q(sqrt(-n))K=\mathbb{Q}(\sqrt{-n})K=Q(−n), let OOO\mathcal{O}O be its ring of integers, and HHHHH its Hilbert class field. Let ccccc be the complex conjugation and let E(K)c=−1⊂E(K)E(K)c=−1⊂E(K)E(K)^(c=-1)sub E(K)E(K)^{c=-1} \subset E(K)E(K)c=−1⊂E(K) be the subgroup on which ccccc acts by -1 , then we naturally have E(K)c=−1≃E(n)(Q)E(K)c=−1≃E(n)(Q)E(K)^(c=-1)≃E^((n))(Q)E(K)^{c=-1} \simeq E^{(n)}(\mathbb{Q})E(K)c=−1≃E(n)(Q).
Define the Heegner point, which lies in E(n)(Q)⊗QE(n)(Q)⊗QE^((n))(Q)oxQE^{(n)}(\mathbb{Q}) \otimes \mathbb{Q}E(n)(Q)⊗Q, as follows:
Here ϵϵepsilon\epsilonϵ is an integer such that ϵ2≡−n(mod128)ϵ2≡−n(mod128)epsilon^(2)-=-n(mod 128)\epsilon^{2} \equiv-n(\bmod 128)ϵ2≡−n(mod128). The construction is natural from an automorphic representation point of view, which will be described in Section 2.
Conjecture B2. Among all positive integers n≡5,6,7(mod8)n≡5,6,7(mod8)n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8), the subset of nnnnn with ynyny_(n)y_{n}yn being nontorsion has density one.
The Gross-Zagier formula (see Section 2) implies that ynyny_(n)y_{n}yn is nontorsion if and only if L′(1,E(n))≠0L′1,E(n)≠0L^(')(1,E^((n)))!=0L^{\prime}\left(1, E^{(n)}\right) \neq 0L′(1,E(n))≠0. Furthermore, Kolyvagin's work shows that if ynyny_(n)y_{n}yn is nontorsion, then the rank of E(n)(Q)E(n)(Q)E^((n))(Q)E^{(n)}(\mathbb{Q})E(n)(Q) is one [36]. The Gross-Zagier formula also helps compute ynyny_(n)y_{n}yn and therefore a generator of E(n)(Q)[19]E(n)(Q)[19]E^((n))(Q)[19]E^{(n)}(\mathbb{Q})[19]E(n)(Q)[19].
Remark 5. The combination of Conjectures B1 and B2B2B2B 2B2 is equivalent to Goldfeld conjecture for congruent elliptic curves, which would solve the CNP for integers with probability one.
Example 1. For n=101,102n=101,102n=101,102n=101,102n=101,102, and 103, the Heegner point ynyny_(n)y_{n}yn is given by
Heegner [28] in 1952 showed that any prime or double prime n≡5,6,7(mod8)n≡5,6,7(mod8)n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8) is a congruent number. Later on, based on Heegner's method, Monsky [40] in 1990 proved that for (p1,p2)≡(1,5)(mod8)(resp.(p1,p2)≡(1,7)(mod8))p1,p2≡(1,5)(mod8)resp.p1,p2≡(1,7)(mod8)(p_(1),p_(2))-=(1,5)(mod8)(resp.(p_(1),p_(2))-=(1,7)(mod8))\left(p_{1}, p_{2}\right) \equiv(1,5)(\bmod 8)\left(\operatorname{resp} .\left(p_{1}, p_{2}\right) \equiv(1,7)(\bmod 8)\right)(p1,p2)≡(1,5)(mod8)(resp.(p1,p2)≡(1,7)(mod8)), two primes such that (p1p2)=−1p1p2=−1((p_(1))/(p_(2)))=-1\left(\frac{p_{1}}{p_{2}}\right)=-1(p1p2)=−1, the product p1p2p1p2p_(1)p_(2)p_{1} p_{2}p1p2 (resp. 2p1p22p1p22p_(1)p_(2)2 p_{1} p_{2}2p1p2 ) is a congruent number. A natural question is to seek congruent numbers with many prime factors. The following was first conjectured by Monsky in [40].
Theorem 6 (Tian [50]). Let nnnnn be the product of an odd number of primes ≡5(mod8)≡5(mod8)-=5(mod8)\equiv 5(\bmod 8)≡5(mod8) that are not quadratic residues to each other, then nnnnn is a congruent number.
Theorem 7 (Burungale-Tian [11]). Conjecture B1 is true, namely among all square-free positive integers n≡1,2,3(mod8)n≡1,2,3(mod8)n-=1,2,3(mod8)n \equiv 1,2,3(\bmod 8)n≡1,2,3(mod8), the subset of nnnnn with L(n)≠0L(n)≠0L(n)!=0\mathscr{L}(n) \neq 0L(n)≠0 has density one. In particular, the density of noncongruent numbers among square-free positive integers ≡1,2,3(mod8)≡1,2,3(mod8)-=1,2,3(mod8)\equiv 1,2,3(\bmod 8)≡1,2,3(mod8) is one.
Let SSSSS be the subset of positive square-free integers n≡5,6,7(mod8)n≡5,6,7(mod8)n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8) so that dimF2Sel2(E(n)/Q)/E(Q)[2]=1dimF2â¡Sel2â¡E(n)/Q/E(Q)[2]=1dim_(F_(2))Sel_(2)(E^((n))//Q)//E(Q)[2]=1\operatorname{dim}_{\mathbb{F}_{2}} \operatorname{Sel}_{2}\left(E^{(n)} / \mathbb{Q}\right) / E(\mathbb{Q})[2]=1dimF2â¡Sel2â¡(E(n)/Q)/E(Q)[2]=1. By the results on distribution of 2-Selmer groups [27, 32,49], the density of SSSSS for all the positive square-free integers n≡5,6,7(mod8)n≡5,6,7(mod8)n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8) is
Theorem 8([47,50,51])8([47,50,51])8([47,50,51])8([47,50,51])8([47,50,51]). There is a density- 2323(2)/(3)\frac{2}{3}23 subset of SSSSS so that the analytic rank of E(n)E(n)E^((n))E^{(n)}E(n) is one and the 2-part BSD formula holds. In particular, among all the square-free positive integers n≡5,6,7(mod8)n≡5,6,7(mod8)n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8), the density of congruent numbers is greater than
The strategy of the proof of Theorems 6 and 8 (resp. Theorem 7) will be given in Section 2 (resp. Section 3).
2. HEEGNER POINT AND EXPLICIT GROSS-ZAGIER FORMULA
Heegner points and Gross-Zagier formula play an important role in the study of elliptic curves. The work of Yuan, Zhang, and Zhang [55] gives the general construction of Heegner points on Shimura curves over totally real fields and establishes the general GrossZagier formula. Some arithmetic applications require an explicit form of the formula such as that in [24]. In this section, we introduce the explicit Gross-Zagier formula from [15] and its application to CNP.
We assume as given:
AAAAA-an elliptic curve over QQQ\mathbb{Q}Q with conductor NNNNN,
KKKKK-an imaginary quadratic field with discriminant DDDDD,
χχchi\chiχ-a ring class character of KKKKK with conductor ccccc, which can be viewed as a character on Gal(Hc/K)Galâ¡Hc/KGal(H_(c)//K)\operatorname{Gal}\left(H_{c} / K\right)Galâ¡(Hc/K), where HcHcH_(c)H_{c}Hc is the ring class field of KKKKK with conductor ccccc,
characterized by the reciprocity law
Here, Oc=Z+cOKOc=Z+cOKO_(c)=Z+cO_(K)\mathcal{O}_{c}=\mathbb{Z}+c \mathcal{O}_{K}Oc=Z+cOK is an order of OKOKO_(K)\mathcal{O}_{K}OK. For any abelian group MMMMM, denote M^=M⊗ZZ^M^=M⊗ZZ^hat(M)=Mox_(Z) hat(Z)\hat{M}=M \otimes_{\mathbb{Z}} \hat{\mathbb{Z}}M^=M⊗ZZ^ with Z^=∏pZpZ^=âˆp Zphat(Z)=prod_(p)Z_(p)\hat{\mathbb{Z}}=\prod_{p} \mathbb{Z}_{p}Z^=âˆpZp.
Assume that the Rankin-Selberg L-series L(s,A,χ)L(s,A,χ)L(s,A,chi)L(s, A, \chi)L(s,A,χ) associated to (A,χ)(A,χ)(A,chi)(A, \chi)(A,χ) has sign -1 in its function equation.
In the following, we shall introduce the construction of the Heegner points and the explicit Gross-Zagier formula for (A,χ)(A,χ)(A,chi)(A, \chi)(A,χ) under the assumption (c,N)=1(c,N)=1(c,N)=1(c, N)=1(c,N)=1. Let BBBBB be the unique indefinite quaternion algebra over QQQ\mathbb{Q}Q whose ramified places are given by all ppppp such that
Denote by XR^×XR^×X_( hat(R)^(xx))X_{\hat{R}^{\times}}XR^×the Shimura curve over QQQ\mathbb{Q}Q associated to BBBBB of level R^×R^×hat(R)^(xx)\hat{R}^{\times}R^×. Under an isomorphism B(R)≃M2(R)B(R)≃M2(R)B(R)≃M_(2)(R)B(\mathbb{R}) \simeq M_{2}(\mathbb{R})B(R)≃M2(R), it has the following complex uniformization:
Denote by [z,g]R^×[z,g]R^×[z,g]_( hat(R)^(xx))[z, g]_{\hat{R}^{\times}}[z,g]R^×the image of (z,g)∈H±×B^×(z,g)∈H±×B^×(z,g)inH^(+-)xx hat(B)^(xx)(z, g) \in \mathscr{H}^{ \pm} \times \hat{B}^{\times}(z,g)∈H±×B^×in XR^×(C)XR^×(C)X_( hat(R)^(xx))(C)X_{\hat{R}^{\times}}(\mathbb{C})XR^×(C). Let ξR^×∈Pic(XR^×)⊗QξR^×∈Picâ¡XR^×⊗Qxi_( hat(R)^(xx))in Pic(X_( hat(R)^(xx)))oxQ\xi_{\hat{R}^{\times}} \in \operatorname{Pic}\left(X_{\hat{R}^{\times}}\right) \otimes \mathbb{Q}ξR^×∈Picâ¡(XR^×)⊗Q be the normalized Hodge class with degree 1 on each connected component of XR^×,Q¯XR^×,Q¯X_( hat(R)^(xx), bar(Q))X_{\hat{R}^{\times}, \overline{\mathbb{Q}}}XR^×,Q¯ (see [55[55[55[55[55, SECTION 3.1.3]).
The following proposition follows from the modularity theorem and the JacquetLanglands correspondence.
Proposition 9 ([15, PROPOSITION 3.8]). Up to scalars, there is a unique nonconstant morphism f:XR^×→f:XR^×→f:X_( hat(R)^(xx))rarrf: X_{\hat{R}^{\times}} \rightarrowf:XR^×→ A over QQQ\mathbb{Q}Q satisfying the following properties:
fffff sends ξR^×ξR^×xi_( hat(R)^(xx))\xi_{\hat{R}^{\times}}ξR^×to the identity OOOOO of AAAAA in the sense that if ξR^×ξR^×xi_( hat(R)^(xx))\xi_{\hat{R}^{\times}}ξR^×is represented by a divisor ∑nixi∑nixisumn_(i)x_(i)\sum n_{i} x_{i}∑nixi on XR^×,Q¯XR^×,Q¯X_( hat(R)^(xx), bar(Q))X_{\hat{R}^{\times}, \overline{\mathbb{Q}}}XR^×,Q¯, then ∑nif(xi)=O∑nifxi=Osumn_(i)f(x_(i))=O\sum n_{i} f\left(x_{i}\right)=O∑nif(xi)=O.
For each place p∣(N,D)p∣(N,D)p∣(N,D)p \mid(N, D)p∣(N,D),
Here, TϖpTÏ–pT_(Ï–_(p))T_{\varpi_{p}}TÏ–p is the automorphism of XR^×XR^×X_( hat(R)^(xx))X_{\hat{R}^{\times}}XR^×, which on XR^×(C)XR^×(C)X_( hat(R)^(xx))(C)X_{\hat{R}^{\times}}(\mathbb{C})XR^×(C) is given by [z,g]R^×↦[z,gϖp]R^×[z,g]R^×↦z,gÏ–pR^×[z,g]_( hat(R)^(xx))|->[z,gÏ–_(p)]_( hat(R)^(xx))[z, g]_{\hat{R}^{\times}} \mapsto\left[z, g \varpi_{p}\right]_{\hat{R}^{\times}}[z,g]R^×↦[z,gÏ–p]R^×, with ϖp∈Kp×Ï–p∈Kp×ϖ_(p)inK_(p)^(xx)\varpi_{p} \in K_{p}^{\times}Ï–p∈Kp×being any uniformizer of KpKpK_(p)K_{p}Kp.
Let z∈Hz∈Hz inHz \in \mathscr{H}z∈H be the unique point fixed by K×K×K^(xx)K^{\times}K×and let P=[z,1]R^×P=[z,1]R^×P=[z,1]_( hat(R)^(xx))P=[z, 1]_{\hat{R}^{\times}}P=[z,1]R^×. By the theory of complex multiplication, PPPPP is defined over the ring class field HcHcH_(c)H_{c}Hc of KKKKK with conductor ccccc and the Galois action is given by
Here, ϕÏ•phi\phiÏ• is the newform associated to AAAAA with
(ϕ,ϕ)Γ0(N)=∬Γ0(N)∖H|ϕ(x+iy)|2dxdy(Ï•,Ï•)Γ0(N)=∬Γ0(N)∖H |Ï•(x+iy)|2dxdy(phi,phi)_(Gamma_(0)(N))=∬_(Gamma_(0)(N)\\H)|phi(x+iy)|^(2)dxdy(\phi, \phi)_{\Gamma_{0}(N)}=\iint_{\Gamma_{0}(N) \backslash \mathscr{H}}|\phi(x+i y)|^{2} d x d y(Ï•,Ï•)Γ0(N)=∬Γ0(N)∖H|Ï•(x+iy)|2dxdy
(1) To compute Heegner points (if non-torsion) via CM theory and modular parameterization, one only gets an approximation. The precise computation can be carried out since one knows the height of Heegner point via the above formula (see [53][53][53][53][53] ).
(2) One may use different Heegner points to construct rational points on AAAAA by choosing different KKKKK. The case (D,N)≠1(D,N)≠1(D,N)!=1(D, N) \neq 1(D,N)≠1 sometimes provides points with smaller height. The above formula with (D,N)≠1(D,N)≠1(D,N)!=1(D, N) \neq 1(D,N)≠1 was conjectured by Gross and Hayashi in [25] and employed in [53] for the computation of rational points.
Some arithmetic problems lead to the situation (c,N)≠1(c,N)≠1(c,N)!=1(c, N) \neq 1(c,N)≠1. Consider the following: a nonzero rational number is called a cube sum if it is of form a3+b3a3+b3a^(3)+b^(3)a^{3}+b^{3}a3+b3 with a,b∈Q×a,b∈Q×a,b inQ^(xx)a, b \in \mathbb{Q}^{\times}a,b∈Q×. For any n∈Q×n∈Q×n inQ^(xx)n \in \mathbb{Q}^{\times}n∈Q×, consider the elliptic curve Cn:x3+y3=2nCn:x3+y3=2nC_(n):x^(3)+y^(3)=2nC_{n}: x^{3}+y^{3}=2 nCn:x3+y3=2n. If nnnnn is not a cube, then 2n2n2n2 n2n is a cube sum if and only if the rank of Cn(Q)Cn(Q)C_(n)(Q)C_{n}(\mathbb{Q})Cn(Q) is positive.
Theorem 12 ([16]). For any odd integer k≥1k≥1k >= 1k \geq 1k≥1, there exist infinitely many cube-free odd integers nnnnn with exactly kkkkk distinct prime factors such that
Here, a certain Heegner point is considered for the pair (A,χ)(A,χ)(A,chi)(A, \chi)(A,χ) where A=X0(36)A=X0(36)A=X_(0)(36)A=X_{0}(36)A=X0(36) : x2=y3+1x2=y3+1x^(2)=y^(3)+1x^{2}=y^{3}+1x2=y3+1 is an isogeny to C1C1C_(1)C_{1}C1 and χχchi\chiχ is a certain cubic ring class character over the imaginary quadratic field Q(−3)Q(−3)Q(sqrt(-3))\mathbb{Q}(\sqrt{-3})Q(−3) with conductor 3n∗3n∗3n^(**)3 n^{*}3n∗ where n∗n∗n^(**)n^{*}n∗ is the product of prime factors in nnnnn. In particular, the pair (A,χ)(A,χ)(A,chi)(A, \chi)(A,χ) has joint ramification at the prime 3 .
In fact, the explicit Gross-Zagier formula is proved for any pair (π,χ)(Ï€,χ)(pi,chi)(\pi, \chi)(Ï€,χ) where
πÏ€pi\piÏ€ is a cuspidal automorphic representation on GL2GL2GL_(2)\mathrm{GL}_{2}GL2 over a totally real field FFFFF with central character ωπωπomega_(pi)\omega_{\pi}ωπ, discrete series of weight 2 at all archimedean places,
χ:K×∖K^×→C×χ:K×∖K^×→C×chi:K^(xx)\\ hat(K)^(xx)rarrC^(xx)\chi: K^{\times} \backslash \hat{K}^{\times} \rightarrow \mathbb{C}^{\times}χ:K×∖K^×→C×is a character of finite order for a totally imaginary quadratic extension KKKKK over FFFFF such that
(ii) the root number of the Rankin-Selberg L-series L(s,π,χ)L(s,Ï€,χ)L(s,pi,chi)L(s, \pi, \chi)L(s,Ï€,χ) is -1 .
Based on the work of Yuan-Zhang-Zhang [55], the above explicit formula is established via generalizing Gross-Prasad local test vector theory.
The relevant problem in local harmonic analysis is the following. Let BBB\mathscr{B}B be a quaternion algebra over a local field FFF\mathscr{F}F with a quadratic sub- FFF\mathcal{F}F-algebra KKK\mathcal{K}K. Let πÏ€pi\piÏ€ be an irreducible smooth admissible representation on B×B×B^(xx)\mathscr{B}^{\times}B×which is of infinite dimension if BBB\mathscr{B}B is split. Let χχchi\chiχ
In general, dimCP(π,χ)≤1dimCâ¡P(Ï€,χ)≤1dim_(C)P(pi,chi) <= 1\operatorname{dim}_{\mathbb{C}} \mathcal{P}(\pi, \chi) \leq 1dimCâ¡P(Ï€,χ)≤1. In the case P(π,χ)≠0P(Ï€,χ)≠0P(pi,chi)!=0\mathcal{P}(\pi, \chi) \neq 0P(Ï€,χ)≠0, a vector φφvarphi\varphiφ is called a test vector for (π,χ)(Ï€,χ)(pi,chi)(\pi, \chi)(Ï€,χ) if ℓ(φ)≠0â„“(φ)≠0â„“(varphi)!=0\ell(\varphi) \neq 0â„“(φ)≠0 for any nonzero ℓ∈P(π,χ)ℓ∈P(Ï€,χ)â„“inP(pi,chi)\ell \in \mathcal{P}(\pi, \chi)ℓ∈P(Ï€,χ).
Moreover, for an unitary (π,χ),P(π,χ)(Ï€,χ),P(Ï€,χ)(pi,chi),P(pi,chi)(\pi, \chi), \mathscr{P}(\pi, \chi)(Ï€,χ),P(Ï€,χ) is nonzero if and only if the bilinear form α∈P(π,χ)⊗P(π¯,χ¯)α∈P(Ï€,χ)⊗P(π¯,χ¯)alpha inP(pi,chi)oxP( bar(pi), bar(chi))\alpha \in \mathcal{P}(\pi, \chi) \otimes \mathscr{P}(\bar{\pi}, \bar{\chi})α∈P(Ï€,χ)⊗P(π¯,χ¯) defined as the toric integral of matrix coefficients
For any pair (π,χ)(Ï€,χ)(pi,chi)(\pi, \chi)(Ï€,χ) as above, in [15] we find an admissible order RRR\mathcal{R}R for (π,χ)(Ï€,χ)(pi,chi)(\pi, \chi)(Ï€,χ) which is unique up to K×K×K^(xx)\mathcal{K}^{\times}K×-conjugacy. The invariant subspace πR×Ï€R×pi^(R^(xx))\pi^{\mathcal{R}^{\times}}Ï€R×of πÏ€pi\piÏ€ by R×R×R^(xx)\mathcal{R}^{\times}R×is at most of dimension 2. By studying the toric integral ααalpha\alphaα, there is a line in πR×Ï€R×pi^(R^(xx))\pi^{\mathcal{R}^{\times}}Ï€R×containing test vectors for (π,χ)(Ï€,χ)(pi,chi)(\pi, \chi)(Ï€,χ).
Our explicit Gross-Zagier formula satisfies the following properties: First, the test vector only depends on the local type πv,χvÏ€v,χvpi_(v),chi_(v)\pi_{v}, \chi_{v}Ï€v,χv, for vvvvv dividing the conductor of πÏ€pi\piÏ€. It is useful when considering horizontal variation (quadratic twist, for example), see [7,13], or vertical variation (in Iwasawa theory) of the character χχchi\chiχ. We also have a so-called SSSSS-version formula which says that for a different choice of a pure tensor test vector, for example, at a finite set of places SSSSS, the new explicit formula can be obtained by modifying the original one by local computations at SSSSS, for example, see [16].
In the rest of this section, we sketch a proofs of Theorems 6 and 8. In Heegner's work, the point ynyny_(n)y_{n}yn is not 2-divisible. In [50,51], Heegner's results were generalized to many prime factors by induction on 2-divisibility of Heegner points via the Waldspurger and GrossZagier formulas.
For E:y2=x3−x,K=Q(−n),n≡5,6,7E:y2=x3−x,K=Q(−n),n≡5,6,7E:y^(2)=x^(3)-x,K=Q(sqrt(-n)),n-=5,6,7E: y^{2}=x^{3}-x, K=\mathbb{Q}(\sqrt{-n}), n \equiv 5,6,7E:y2=x3−x,K=Q(−n),n≡5,6,7 positive square-free, the explicit Gross-Zagier formula for (E,K)(E,K)(E,K)(E, K)(E,K) gives
where a=1,0,1a=1,0,1a=1,0,1a=1,0,1a=1,0,1 in the case n≡5,6,7(mod8)n≡5,6,7(mod8)n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8), respectively. Now if ynyny_(n)y_{n}yn is nontorsion, then the BSD conjecture becomes
where μ(n)μ(n)mu(n)\mu(n)μ(n) is the number of odd prime factors of nnnnn. If nnnnn is a prime then the 2-part of the BSD conjecture is equivalent to 2-indivisibility of ynyny_(n)y_{n}yn, this is exactly Heegner's case. As μ(n)μ(n)mu(n)\mu(n)μ(n) becomes large, the 2 divisibility of ynyny_(n)y_{n}yn becomes high and the original Heegner's argument does not work directly. Whenever dimF2Sel2(E(n)/Q)/E(n)(Q)[2]=1dimF2â¡Sel2â¡E(n)/Q/E(n)(Q)[2]=1dim_(F_(2))Sel_(2)(E^((n))//Q)//E^((n))(Q)[2]=1\operatorname{dim}_{\mathbb{F}_{2}} \operatorname{Sel}_{2}\left(E^{(n)} / \mathbb{Q}\right) / E^{(n)}(\mathbb{Q})[2]=1dimF2â¡Sel2â¡(E(n)/Q)/E(n)(Q)[2]=1, the 2-divisibility of ynyny_(n)y_{n}yn fully comes from Tamagawa numbers. The 2-divisibility can be proved via induction; to do this, one employs various relations between different Heegner points.
We employ the induction method (see [50]) in the case n≡5(mod8)n≡5(mod8)n-=5(mod8)n \equiv 5(\bmod 8)n≡5(mod8) with all prime factors ≡1(mod4)≡1(mod4)-=1(mod4)\equiv 1(\bmod 4)≡1(mod4). Let zn:=f(τn)zn:=fÏ„nz_(n):=f(tau_(n))z_{n}:=f\left(\tau_{n}\right)zn:=f(Ï„n) and let ynyny_(n)y_{n}yn be the Heegner points as in the Section 1. Denote by HHHHH the Hilbert class field of K=Q(−n)K=Q(−n)K=Q(sqrt(-n))K=\mathbb{Q}(\sqrt{-n})K=Q(−n) and let H0⊂HH0⊂HH_(0)sub HH_{0} \subset HH0⊂H be the genus subfield determined by Gal(H0/K)≃2Cl(K)Galâ¡H0/K≃2Cl(K)Gal(H_(0)//K)≃2Cl(K)\operatorname{Gal}\left(H_{0} / K\right) \simeq 2 \mathrm{Cl}(K)Galâ¡(H0/K)≃2Cl(K). For each d∣nd∣nd∣nd \mid nd∣n with the same above property as nnnnn, let y0=trH/H0zny0=trH/H0â¡zny_(0)=tr_(H//H_(0))z_(n)y_{0}=\operatorname{tr}_{H / H_{0}} z_{n}y0=trH/H0â¡zn and yd,0=trH/K(−d)znyd,0=trH/K(−d)â¡zny_(d,0)=tr_(H//K(sqrt(-d)))z_(n)y_{d, 0}=\operatorname{tr}_{H / K(\sqrt{-d})} z_{n}yd,0=trH/K(−d)â¡zn. Then these points satisfy the following relation:
yn+∑1≤d∣n,d≠n,d≡5(mod8)yd,0=2μ(n)y0(modE[2])yn+∑1≤d∣n,d≠n,d≡5(mod8) yd,0=2μ(n)y0(modE[2])y_(n)+sum_({:1 <= d∣n","d!=n","d-=5(mod8):})y_(d,0)=2^(mu(n))y_(0)quad(mod E[2])y_{n}+\sum_{\substack{1 \leq d \mid n, d \neq n, d \equiv 5(\bmod 8)}} y_{d, 0}=2^{\mu(n)} y_{0} \quad(\bmod E[2])yn+∑1≤d∣n,d≠n,d≡5(mod8)yd,0=2μ(n)y0(modE[2])
Furthermore, the Gross-Zagier formula implies that whenever yd,0yd,0y_(d,0)y_{d, 0}yd,0 is nontorsion, both yd,0yd,0y_(d,0)y_{d, 0}yd,0 and ydydy_(d)y_{d}yd lie in the one-dimensional space E(Q(−d))c=−1⊗QE(Q(−d))c=−1⊗QE(Q(sqrt(-d)))^(c=-1)oxQE(\mathbb{Q}(\sqrt{-d}))^{c=-1} \otimes \mathbb{Q}E(Q(−d))c=−1⊗Q and
By induction on the 2 divisibility of ydydy_(d)y_{d}yd and 2 divisibility of Lals (1,E(n/d))Lalg(1,E)Lals 1,E(n/d)Lalg(1,E)(L^("als ")(1,E^((n//d))))/(L^(alg)(1,E))\frac{L^{\text {als }}\left(1, E^{(n / d)}\right)}{L^{\mathrm{alg}}(1, E)}Lals (1,E(n/d))Lalg(1,E), one gets the following 2 divisibility of ynyny_(n)y_{n}yn whenever Cl(K)Cl(K)Cl(K)\mathrm{Cl}(K)Cl(K) has no element of order 4 :
The above induction argument was improved in [51] to handle the general case. For a positive integer ddddd, let g(d)=#2Cl(Q(−d))g(d)=#2Cl(Q(−d))g(d)=#2Cl(Q(sqrt(-d)))g(d)=\# 2 \mathrm{Cl}(\mathbb{Q}(\sqrt{-d}))g(d)=#2Cl(Q(−d)) be the genus class number of Q(−d)Q(−d)Q(sqrt(-d))\mathbb{Q}(\sqrt{-d})Q(−d). For n≡5,6,7(mod8)n≡5,6,7(mod8)n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8), let
where a(n)=0a(n)=0a(n)=0a(n)=0a(n)=0 if nnnnn is even and a(n)=1a(n)=1a(n)=1a(n)=1a(n)=1 if nnnnn is odd.
Then the BSD conjecture for E(n)E(n)E^((n))E^{(n)}E(n) is equivalent to L(n)2=#⨿(E(n)/Q)L(n)2=#⨿E(n)/QL(n)^(2)=#⨿(E^((n))//Q)\mathscr{L}(n)^{2}=\# \amalg\left(E^{(n)} / \mathbb{Q}\right)L(n)2=#⨿(E(n)/Q) whenever L′(1,E(n))≠0L′1,E(n)≠0L^(')(1,E^((n)))!=0L^{\prime}\left(1, E^{(n)}\right) \neq 0L′(1,E(n))≠0. We have the following criterion for the 2-indivisibility of L(n)L(n)L(n)\mathscr{L}(n)L(n) :
Theorem 13 ([51]). For n≡5,6,7(mod8)n≡5,6,7(mod8)n-=5,6,7(mod8)n \equiv 5,6,7(\bmod 8)n≡5,6,7(mod8) positive square-free, L(n)L(n)L(n)\mathscr{L}(n)L(n) is an integer and 2−ρ(n)L(n)2−Ï(n)L(n)2^(-rho(n))L(n)2^{-\rho(n)} \mathscr{L}(n)2−Ï(n)L(n) is odd if
where ρ(n)Ï(n)rho(n)\rho(n)Ï(n) is a positive integer (defined in [51]) arising from an isogeny between E(n)E(n)E^((n))E^{(n)}E(n) and 2ny2=x3+x2ny2=x3+x2ny^(2)=x^(3)+x2 n y^{2}=x^{3}+x2ny2=x3+x.
Consider the following sets for i=5,6,7i=5,6,7i=5,6,7i=5,6,7i=5,6,7 :
ΣiΣiSigma_(i)\Sigma_{i}Σi-the set of all square-free positive integers n≡i(mod8)n≡i(mod8)n-=i(mod8)n \equiv i(\bmod 8)n≡i(mod8),
Σi′⊂ΣiΣi′⊂ΣiSigma_(i)^(')subSigma_(i)\Sigma_{i}^{\prime} \subset \Sigma_{i}Σi′⊂Σi-the subset of nnnnn with s(n)=1s(n)=1s(n)=1s(n)=1s(n)=1,
Σi′′⊂ΣiΣi′′⊂ΣiSigma_(i)^('')subSigma_(i)\Sigma_{i}^{\prime \prime} \subset \Sigma_{i}Σi′′⊂Σi-the subset of nnnnn satisfying the conditions in the Theorem 13.
Theorem 14 (Heath-Brown [27], Swinnerton-Dyer [49], Kane [32]). The density of Σi′Σi′Sigma_(i)^(')\Sigma_{i}^{\prime}Σi′ in ΣiΣiSigma_(i)\Sigma_{i}Σi is
Theorem 15 (Smith [47]). The set Σi′′Σi′′Sigma_(i)^('')\Sigma_{i}^{\prime \prime}Σi′′ is contained in Σi′Σi′Sigma_(i)^(')\Sigma_{i}^{\prime}Σi′ with density 34,12,3434,12,34(3)/(4),(1)/(2),(3)/(4)\frac{3}{4}, \frac{1}{2}, \frac{3}{4}34,12,34 for i=5,6,7i=5,6,7i=5,6,7i=5,6,7i=5,6,7, respectively.
Observe that Theorem 8 is a consequence of Theorems 13, 14, and 15.
3. SELMER GROUPS: P-CONVERSE AND DISTRIBUTION
The nnnnn-Selmer group for an elliptic curve AAAAA over a number FFFFF is defined by
The group Hom(Selp∞(A/F),Qp/Zp)Homâ¡Selpâ¡âˆž(A/F),Qp/ZpHom(Sel_(p)oo(A//F),Q_(p)//Z_(p))\operatorname{Hom}\left(\operatorname{Sel}_{p} \infty(A / F), \mathbb{Q}_{p} / \mathbb{Z}_{p}\right)Homâ¡(Selpâ¡âˆž(A/F),Qp/Zp) is known to be a finitely generated ZpZpZ_(p)\mathbb{Z}_{p}Zp-module, its rank of free part is called p∞p∞p^(oo)p^{\infty}p∞-Selmer corank of AAAAA, denoted by corank ZpSelp∞(A/F)ZpSelp∞â¡(A/F)Z_(p)Sel_(p^(oo))(A//F)\mathbb{Z}_{p} \operatorname{Sel}_{p^{\infty}}(A / F)ZpSelp∞â¡(A/F).
Conjecture 16 (BSD, reformulation). Let A/FA/FA//FA / FA/F be an elliptic curve over a number field, r∈Z>0r∈Z>0r inZ_( > 0)r \in \mathbb{Z}_{>0}r∈Z>0, and ppppp be a prime. Then the following are equivalent:
(1) ords=1L(s,A/F)=rords=1â¡L(s,A/F)=rord_(s=1)L(s,A//F)=r\operatorname{ord}_{s=1} L(s, A / F)=rords=1â¡L(s,A/F)=r,
(2) rankZA(F)=rrankZâ¡A(F)=rrank_(Z)A(F)=r\operatorname{rank}_{\mathbb{Z}} A(F)=rrankZâ¡A(F)=r and ⨿(A/F)⨿(A/F)⨿(A//F)\amalg(A / F)⨿(A/F) is finite,
We can also consider a Selmer variant of Goldfeld's Conjecture 3. The following was conjectured by Bhargava-Kane-Lenstra-Poonen-Rains.
Conjecture 17 ([4]). Let FFFFF be a global field, ppppp be a prime, and GGGGG a finite symplectic ppppp-group. If all elliptic curves A over FFFFF are ordered by height, then for r=0,1r=0,1r=0,1r=0,1r=0,1 we have
In particular, the density of rank 0 (or 1 ) elliptic curves over FFFFF is 1212(1)/(2)\frac{1}{2}12.
3.1. Distribution of Selmer groups and Goldfeld conjecture
The following Smith's result shows that even for a quadratic twist family, the distribution follows the same pattern as the above conjecture.
Theorem 18 (Smith [48]). Let A/QA/QA//QA / \mathbb{Q}A/Q be an elliptic curve satisfying AAAAA has full rational 2 torsion, but no rational cyclic subgroup of order 4 . Then, among the quadratic twists A(d)A(d)A^((d))A^{(d)}A(d) of A, a distribution law as in Conjecture 17 holds for p=2p=2p=2p=2p=2.
In particular, among all quadratic twists A(d)A(d)A^((d))A^{(d)}A(d) with sign +1 (resp. -1 ), there is a subset of density one with corank Z2Sel2(A(d)/Q)=0Z2Sel2â¡A(d)/Q=0Z_(2)Sel_(2)(A^((d))//Q)=0\mathbb{Z}_{2} \operatorname{Sel}_{2}\left(A^{(d)} / \mathbb{Q}\right)=0Z2Sel2â¡(A(d)/Q)=0 (resp. 1 ).
Remark 19. Smith's work is based on the following results of Heath-Brown, SwinnertonDyer, and Kane on distribution of 2-Selmer groups, which is the first step to understand the distribution of 2∞2∞2^(oo)2^{\infty}2∞-Selmer groups.
Theorem 20 ([27,32,49]). Let A/QA/QA//QA / \mathbb{Q}A/Q be an elliptic curve satisfying
A has full rational 2-torsion, but no rational cyclic subgroup of order 4 .
Then for r∈Z≥0r∈Z≥0r inZ_( >= 0)r \in \mathbb{Z}_{\geq 0}r∈Z≥0, among the quadratic twists A(d)A(d)A^((d))A^{(d)}A(d) of AAAAA,
In general, for a quadratic twist family of elliptic curves over QQQ\mathbb{Q}Q, its distribution of 2-Selmer groups may exhibit new behavior. For example, the quadratic twist family of Tiling
number elliptic curves has
A(d):dy2=x(x−1)(x+3) with A(1)(Q)tor ≅Z/2Z×Z/4ZA(d):dy2=x(x−1)(x+3) with A(1)(Q)tor ≅Z/2Z×Z/4ZA^((d)):dy^(2)=x(x-1)(x+3)quad" with "A^((1))(Q)_("tor ")~=Z//2ZxxZ//4ZA^{(d)}: d y^{2}=x(x-1)(x+3) \quad \text { with } A^{(1)}(\mathbb{Q})_{\text {tor }} \cong \mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 4 \mathbb{Z}A(d):dy2=x(x−1)(x+3) with A(1)(Q)tor ≅Z/2Z×Z/4Z
Perhaps surprisingly, in light of the presence of such rational 4-torsion, the distribution of 2-Selmer groups no longer seems to be as in Theorem 20. For example, if d≠1d≠1d!=1d \neq 1d≠1, d≡1(mod12)d≡1(mod12)d-=1(mod 12)d \equiv 1(\bmod 12)d≡1(mod12) is positive square-free, then
A preliminary study suggests that for such elliptic curves, the distribution of 2Selmer groups may look more like that of the 4-ranks of ideal class groups of the underlying imaginary quadratic fields.
Theorem 21 ([22]). Let AAAAA be the elliptic curve y2=x(x−1)(x+3)y2=x(x−1)(x+3)y^(2)=x(x-1)(x+3)y^{2}=x(x-1)(x+3)y2=x(x−1)(x+3). Among the set of positive square-free integers d≡7(mod24)(d≡7(mod24)(d-=7(mod 24)(d \equiv 7(\bmod 24)(d≡7(mod24)( resp. d≡3(mod24))d≡3(mod24))d-=3(mod 24))d \equiv 3(\bmod 24))d≡3(mod24)), the subset of ddddd such that both of A(±d)A(±d)A^((+-d))A^{( \pm d)}A(±d) have Sel2(A(±d)/Q)/A(±d)(Q)[2]Sel2â¡A(±d)/Q/A(±d)(Q)[2]Sel_(2)(A^((+-d))//Q)//A^((+-d))(Q)[2]\operatorname{Sel}_{2}\left(A^{( \pm d)} / \mathbb{Q}\right) / A^{( \pm d)}(\mathbb{Q})[2]Sel2â¡(A(±d)/Q)/A(±d)(Q)[2] trivial has density 12∏i=1∞(1−2−i)>12âˆi=1∞ 1−2−i>(1)/(2)prod_(i=1)^(oo)(1-2^(-i)) >\frac{1}{2} \prod_{i=1}^{\infty}\left(1-2^{-i}\right)>12âˆi=1∞(1−2−i)>14.4%14.4%14.4%14.4 \%14.4% (resp. of density ∏i=1∞(1−2−i)>28.8%âˆi=1∞ 1−2−i>28.8%prod_(i=1)^(oo)(1-2^(-i)) > 28.8%\prod_{i=1}^{\infty}\left(1-2^{-i}\right)>28.8 \%âˆi=1∞(1−2−i)>28.8% ).
In general, for r∈Z≥0r∈Z≥0r inZ_( >= 0)r \in \mathbb{Z}_{\geq 0}r∈Z≥0, for the set of positive square-free integers d≡3(mod24)d≡3(mod24)d-=3(mod 24)d \equiv 3(\bmod 24)d≡3(mod24),
An approach to Goldfeld conjecture. The Goldfeld conjecture for a quadratic twist family of elliptic curves over QQQ\mathbb{Q}Q is a consequence of the following steps:
(1) Distribution of p∞p∞p^(oo)p^{\infty}p∞-Selmer groups in the quadratic twist family, which should be a certain variant of general distribution law for all elliptic curves in [4].
(2) The rank zero and rank one ppppp-converse.
Proof of Theorem 7. It is a direct consequence of Tunnell's work on quadratic twist L-values of congruent elliptic curves [52], Theorem 18 of Smith on distribution of 2∞2∞2^(oo)2^{\infty}2∞-Selmer groups, and Theorem 22 below on the rank zero ppppp-converse for CMCMCM\mathrm{CM}CM elliptic curves for p=2p=2p=2p=2p=2.
3.2. Recent progress: ppppp-converse
In the remaining part of this section, we discuss the ppppp-converse theorem in the CMCMCM\mathrm{CM}CM case. For a few other ppppp-converse theorems, see [6,8-10]. Fix a prime ppppp.
Theorem 22 (Rubin [43,44], Burungale-Tian [11]). Let A/QA/QA//QA / \mathbb{Q}A/Q be a CM elliptic curve. Then,
Remark 23. Assume that A/QA/QA//QA / \mathbb{Q}A/Q has CMCMCMC MCM by KKKKK and p∤#OK×p∤#OK×p∤#O_(K)^(xx)p \nmid \# \mathcal{O}_{K}^{\times}p∤#OK×. Then the above theorem is due to Rubin [43,44][43,44][43,44][43,44][43,44].
Remark 24. Skinner-Urban [46] established the rank zero p-converse for certain elliptic curves over QQQ\mathbb{Q}Q without CM.
Theorem 25 (W. Zhang [56], Skinner [45], Castella-Wan [17]). Let A/QQ be a non-CM elliptic curve and p≥3p≥3p >= 3p \geq 3p≥3. Then,
Their methods essentially excludes the CM case. For CM elliptic curves:
Theorem 26 (Burungale-Tian [12], Burungale-Skinner-Tian [8]). Let A be a CM elliptic curve over QQQ\mathbb{Q}Q and p∤6NAp∤6NAp∤6N_(A)p \nmid 6 N_{A}p∤6NA a prime. Then
We outline the proof of Theorem 22. Unconventionally for the CM elliptic curves, this approach is based on Kato's main conjecture [33], which we recall now.
Let f∈Sk(Γ0(N))f∈SkΓ0(N)f inS_(k)(Gamma_(0)(N))f \in S_{k}\left(\Gamma_{0}(N)\right)f∈Sk(Γ0(N)) be an elliptic newform of even weight k≥2k≥2k >= 2k \geq 2k≥2, level Γ0(N)Γ0(N)Gamma_(0)(N)\Gamma_{0}(N)Γ0(N), and Hecke field FFFFF. Fix an embedding ιp:Q→Q¯pιp:Q→Q¯piota_(p):Qrarr bar(Q)_(p)\iota_{p}: \mathbb{Q} \rightarrow \overline{\mathbb{Q}}_{p}ιp:Q→Q¯p. Let λλlambda\lambdaλ be the place of FFFFF induced by ιp,Fλιp,Fλiota_(p),F_(lambda)\iota_{p}, F_{\lambda}ιp,Fλ be the completion of FFFFF at λλlambda\lambdaλ, and OλOλO_(lambda)O_{\lambda}Oλ the integer ring. Let VFλ(f)VFλ(f)V_(F_(lambda))(f)V_{F_{\lambda}}(f)VFλ(f) be the two-dimensional representation of GQGQG_(Q)G_{\mathbb{Q}}GQ over FλFλF_(lambda)F_{\lambda}Fλ associated to fffff. We first introduce the related Iwasawa cohomology. For n∈Z≥0n∈Z≥0n inZ_( >= 0)n \in \mathbb{Z}_{\geq 0}n∈Z≥0, let
Let Λ=Oλ[[G∞]]Λ=OλG∞Lambda=O_(lambda)[[G_(oo)]]\Lambda=O_{\lambda}\left[\left[G_{\infty}\right]\right]Λ=Oλ[[G∞]] be a two-dimensional complete semilocal ring. For q∈Z≥0q∈Z≥0q inZ_( >= 0)q \in \mathbb{Z}_{\geq 0}q∈Z≥0, consider the ΛQp=Λ⊗QΛQp=Λ⊗QLambda_(Q_(p))=Lambda oxQ\Lambda_{\mathbb{Q}_{p}}=\Lambda \otimes \mathbb{Q}ΛQp=Λ⊗Q-module
(1) H2(VFλ(f))H2VFλ(f)H^(2)(V_(F_(lambda))(f))\mathbb{H}^{2}\left(V_{F_{\lambda}}(f)\right)H2(VFλ(f)) is a torsion ΛQpΛQpLambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-module, and
(2) H1(VFλ(f))H1VFλ(f)H^(1)(V_(F_(lambda))(f))\mathbb{H}^{1}\left(V_{F_{\lambda}}(f)\right)H1(VFλ(f)) is a free ΛQpΛQpLambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-module of rank one.
Now we introduce the submodule of H1(VFλ(f))H1VFλ(f)H^(1)(V_(F_(lambda))(f))\mathbb{H}^{1}\left(V_{F_{\lambda}}(f)\right)H1(VFλ(f)) generated by Beilinson-Kato elements. We have the following existence of zeta elements for the ppppp-adic Galois representation corresponding to an elliptic newform [33, Ñ‚HM. 12.5]:
(1) There exists a nonzero FλFλF_(lambda)F_{\lambda}Fλ-linear morphism
(2) Let Z(f)Z(f)Z(f)Z(f)Z(f) be the ΛQpΛQpLambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-submodule of H1(VFλ(f))H1VFλ(f)H^(1)(V_(F_(lambda))(f))\mathbb{H}^{1}\left(V_{F_{\lambda}}(f)\right)H1(VFλ(f)) generated by zγ(f)zγ(f)z_(gamma)(f)z_{\gamma}(f)zγ(f) for all γ∈VFλ(f)γ∈VFλ(f)gamma inV_(F_(lambda))(f)\gamma \in V_{F_{\lambda}}(f)γ∈VFλ(f). Then H1(VFλ(f))/Z(f)H1VFλ(f)/Z(f)H^(1)(V_(F_(lambda))(f))//Z(f)\mathbb{H}^{1}\left(V_{F_{\lambda}}(f)\right) / Z(f)H1(VFλ(f))/Z(f) is a torsion ΛQpΛQpLambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-module.
Remark 27. For a characterizing property of the morphism γ↦zγ(f)γ↦zγ(f)gamma|->z_(gamma)(f)\gamma \mapsto z_{\gamma}(f)γ↦zγ(f) in terms of the underlying critical L-values, we refer to [33, тнм. 12.5 (1)].
Conjecture 28 (Kato's Main Conjecture [33]). The following equality of ideals holds in ΛQpΛQpLambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp :
Theorem 29 (Burungale-Tian [11]). Kato's main conjecture holds for any CM modular form fffff and any prime ppppp.
As observed by Kato [33], the CM case of Kato's main conjecture is closely related to an equivariant main conjecture for the underlying imaginary quadratic field. This is based on an intrinsic relation between the Beilinson-Kato elements and elliptic units.
As a consequence of Theorem 29, we have the following result, which implies Theorem 22 .
Theorem 30 ([11]). Assume that fffff is CMCMCMC MCM. Let Hf1(Q,VFλ(f)(k/2))Hf1Q,VFλ(f)(k/2)H_(f)^(1)(Q,V_(F_(lambda))(f)(k//2))H_{\mathrm{f}}^{1}\left(\mathbb{Q}, V_{F_{\lambda}}(f)(k / 2)\right)Hf1(Q,VFλ(f)(k/2)) be the corresponding Bloch-Kato Selmer group (see Kato [33]). Then,
In the following, we focus on the proof of the rank one CM ppppp-converse theorem for ordinary primes ppppp. The key is an auxiliary Heegner point main conjecture (HPMC, for short).
Classically, HPMC is only formulated for pairs (A,K′)A,K′(A,K^('))\left(A, K^{\prime}\right)(A,K′) where A/QA/QA//QA / \mathbb{Q}A/Q is an elliptic curve and K′K′K^(')K^{\prime}K′ is an imaginary quadratic field satisfying the Heegner hypothesis. To show the rank 1 -converse for a CM elliptic curve, we utilize a certain anticyclotomic Iwasawa theory over the CM field. The key is to construct relevant Heegner points for auxiliary RankinSelberg pairs, and consider the underlying HPMC.
Let A/QA/QA//QA / \mathbb{Q}A/Q be a CM elliptic curve with CMCMCM\mathrm{CM}CM by KKKKK and with p∞p∞p^(oo)p^{\infty}p∞-Selmer corank one. Let λλlambda\lambdaλ be the associated Hecke character over KKKKK and θλθλtheta_(lambda)\theta_{\lambda}θλ the corresponding theta series.
Lemma 31. There exists a finite order Hecke character χχchi\chiχ over KKKKK such that L(1,λ∗/χ∗⋅χ)≠0L1,λ∗/χ∗⋅χ≠0L(1,lambda^(**)//chi^(**)*chi)!=0L\left(1, \lambda^{*} / \chi^{*} \cdot \chi\right) \neq 0L(1,λ∗/χ∗⋅χ)≠0, where ∗∗***∗ is the involution given by nontrivial automorphism of KKKKK, so that the LLLLL-function for the Rankin pair (f:=θλ/χ,χ)f:=θλ/χ,χ(f:=theta_(lambda//chi),chi)\left(f:=\theta_{\lambda / \chi}, \chi\right)(f:=θλ/χ,χ),
L(s,f×χ)=L(s,λ)L(s,λ∗/χ∗⋅χ)L(s,f×χ)=L(s,λ)Ls,λ∗/χ∗⋅χL(s,f xx chi)=L(s,lambda)L(s,lambda^(**)//chi^(**)*chi)L(s, f \times \chi)=L(s, \lambda) L\left(s, \lambda^{*} / \chi^{*} \cdot \chi\right)L(s,f×χ)=L(s,λ)L(s,λ∗/χ∗⋅χ)
has sign -1 and the same vanishing order at the center as L(s,λ)=L(s,A/Q)L(s,λ)=L(s,A/Q)L(s,lambda)=L(s,A//Q)L(s, \lambda)=L(s, A / \mathbb{Q})L(s,λ)=L(s,A/Q).
We have the relevant Heegner point P0∈B(K)⊗QP0∈B(K)⊗QP_(0)in B(K)oxQP_{0} \in B(K) \otimes \mathbb{Q}P0∈B(K)⊗Q on the abelian variety B:=Af×χB:=Af×χB:=A_(f xx chi)B:=A_{f \times \chi}B:=Af×χ associated to the pair (f=θλ/χ,χ)f=θλ/χ,χ(f=theta_(lambda//chi),chi)\left(f=\theta_{\lambda / \chi}, \chi\right)(f=θλ/χ,χ). The Gross-Zagier formula of YuanZhang-Zhang [55] implies that 0≠P0∈B(K)⊗Q0≠P0∈B(K)⊗Q0!=P_(0)in B(K)oxQ0 \neq P_{0} \in B(K) \otimes \mathbb{Q}0≠P0∈B(K)⊗Q if and only if ord s=1L(s,f×χ)=1s=1L(s,f×χ)=1_(s=1)L(s,f xx chi)=1{ }_{s=1} L(s, f \times \chi)=1s=1L(s,f×χ)=1.
Note that ppppp is split in KKKKK. Let K∞/KK∞/KK_(oo)//KK_{\infty} / KK∞/K be the anticyclotomic extension with Galois group Γ≅ZpΓ≅ZpGamma~=Z_(p)\Gamma \cong \mathbb{Z}_{p}Γ≅Zp. For each n≥1n≥1n >= 1n \geq 1n≥1, let Kn⊂K∞Kn⊂K∞K_(n)subK_(oo)K_{n} \subset K_{\infty}Kn⊂K∞ be the degree pnpnp^(n)p^{n}pn subextension over KKKKK.
One can construct a family of norm compatible Heegner points Pn∈B(Kn)Pn∈BKnP_(n)in B(K_(n))P_{n} \in B\left(K_{n}\right)Pn∈B(Kn). Denote by Λ=Op[[Γ]]Λ=Op[[Γ]]Lambda=O_(p)[[Gamma]]\Lambda=\mathcal{O}_{\mathfrak{p}}[[\Gamma]]Λ=Op[[Γ]] and ΛQp=Λ⊗QΛQp=Λ⊗QLambda_(Q_(p))=Lambda oxQ\Lambda_{\mathbb{Q}_{p}}=\Lambda \otimes \mathbb{Q}ΛQp=Λ⊗Q. Here OOO\mathcal{O}O is the endomorphism ring of BBBBB (viewed as a subring of Q¯),p∣pQ¯),p∣pbar(Q)),p∣p\overline{\mathbb{Q}}), \mathfrak{p} \mid pQ¯),p∣p the prime ideal of OOO\mathcal{O}O induced by ιpιpiota_(p)\iota_{p}ιp, and OpOpO_(p)\mathcal{O}_{\mathfrak{p}}Op the completion of OOO\mathcal{O}O at ppppp.
Proposition 32. The ΛQpΛQpLambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-modules
is not ΛQpΛQpLambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-torsion so that S(B/K∞)/ΛQp⋅κSB/K∞/ΛQp⋅κS(B//K_(oo))//Lambda_(Q_(p))*kappaS\left(B / K_{\infty}\right) / \Lambda_{\mathbb{Q}_{p}} \cdot \kappaS(B/K∞)/ΛQp⋅κ is a finitely generated torsion ΛQp-module. ΛQp-module. Lambda_(Q_(p)"-module. ")\Lambda_{\mathbb{Q}_{p} \text {-module. }}ΛQp-module. .
Now corank ZpSelp∞(A/Q)=1ZpSelp∞â¡(A/Q)=1Z_(p)Sel_(p oo)(A//Q)=1\mathbb{Z}_{p} \operatorname{Sel}_{p \infty}(A / \mathbb{Q})=1ZpSelp∞â¡(A/Q)=1 implies that the left-hand side of (∗)(∗)(**)(*)(∗) holds. On the other hand, under the Gross-Zagier formula, ord s=1L(s,A/Q)=1s=1L(s,A/Q)=1_(s=1)L(s,A//Q)=1{ }_{s=1} L(s, A / \mathbb{Q})=1s=1L(s,A/Q)=1 is a consequence of the righthand side of (∗)(∗)(**)(*)(∗).
First proof of HPMC. The two variable Rankin-Selberg ppppp-adic L-function Lp(f×χ)Lp(f×χ)L_(p)(f xx chi)\mathscr{L}_{p}(f \times \chi)Lp(f×χ) (see [21]) associated to (f,χ)(f,χ)(f,chi)(f, \chi)(f,χ) has a decomposition in terms of Lp(λ)Lp(λ)L_(p)(lambda)\mathscr{L}_{p}(\lambda)Lp(λ) and Lp(λ∗/χ∗⋅χ)Lpλ∗/χ∗⋅χL_(p)(lambda^(**)//chi^(**)*chi)\mathscr{L}_{p}\left(\lambda^{*} / \chi^{*} \cdot \chi\right)Lp(λ∗/χ∗⋅χ), where Lp(λ),Lp(λ∗/χ∗⋅χ)Lp(λ),Lpλ∗/χ∗⋅χL_(p)(lambda),L_(p)(lambda^(**)//chi^(**)*chi)\mathscr{L}_{p}(\lambda), \mathscr{L}_{p}\left(\lambda^{*} / \chi^{*} \cdot \chi\right)Lp(λ),Lp(λ∗/χ∗⋅χ) are the Katz ppppp-adic L-functions (see [29]) associated to λλlambda\lambdaλ and λ∗/χ∗λ∗/χ∗lambda^(**)//chi^(**)\lambda^{*} / \chi^{*}λ∗/χ∗. χχchi\chiχ, respectively. Note that Lp(f×χ)Lp(f×χ)L_(p)(f xx chi)\mathscr{L}_{p}(f \times \chi)Lp(f×χ) and Lp(λ)Lp(λ)L_(p)(lambda)\mathscr{L}_{p}(\lambda)Lp(λ) vanish along the anticyclotomic line, thus we may consider their derivatives with respect to the cyclotomic variable, i.e.,
(Lp′(f×χ))=(Lp′(λ))(Lp(λ∗/χ∗⋅χ))Lp′(f×χ)=Lp′(λ)Lpλ∗/χ∗⋅χ(L_(p)^(')(f xx chi))=(L_(p)^(')(lambda))(L_(p)(lambda^(**)//chi^(**)*chi))\left(\mathscr{L}_{p}^{\prime}(f \times \chi)\right)=\left(\mathscr{L}_{p}^{\prime}(\lambda)\right)\left(\mathscr{L}_{p}\left(\lambda^{*} / \chi^{*} \cdot \chi\right)\right)(Lp′(f×χ))=(Lp′(λ))(Lp(λ∗/χ∗⋅χ))
The HPMC is based on ΛΛLambda\LambdaΛ-adic Gross-Zagier formula and Rubin's main conjecture.
On the one hand, the ΛΛLambda\LambdaΛ-adic Gross-Zagier formula [21] connects Heegner point with Lp′(f×χ)Lp′(f×χ)L_(p)^(')(f xx chi)\mathscr{L}_{p}^{\prime}(f \times \chi)Lp′(f×χ) as
where R(λ)R(λ)R(lambda)R(\lambda)R(λ) is the ΛΛLambda\LambdaΛ-adic regulator which is nonzero [5], X(λ)X(λ)X(lambda)X(\lambda)X(λ) and X(λ∗/χ∗⋅χ)Xλ∗/χ∗⋅χX(lambda^(**)//chi^(**)*chi)X\left(\lambda^{*} / \chi^{*} \cdot \chi\right)X(λ∗/χ∗⋅χ) are certain anticyclotomic Selmer groups. Then, the HPMC follows from the decomposition
and the comparison of ΛΛLambda\LambdaΛ-adic regulators, R(λ)=R(f×χ)R(λ)=R(f×χ)R(lambda)=R(f xx chi)R(\lambda)=R(f \times \chi)R(λ)=R(f×χ).
Second proof of HPMC. Via ΛΛLambda\LambdaΛ-adic Waldspurger formula and nontriviality of κκkappa\kappaκ, the HPMC is equivalent to the BDP main conjecture [45]. Let p=vv¯p=vv¯p=v bar(v)p=v \bar{v}p=vv¯ where vvvvv is determined by ιpιpiota_(p)\iota_{p}ιp.
Proposition 34. The ΛQpΛQpLambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-modules
are finitely generated torsion ΛQpΛQpLambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-modules. Here Selpm,vSelpm,vSel_(p^(m),v)\operatorname{Sel}_{p^{m}, v}Selpm,v is the pmpmp^(m)\mathfrak{p}^{m}pm-Selmer group with vvvvv relaxed and v¯v¯bar(v)\bar{v}v¯-strict local Selmer condition [45].
Let Lv(B/K∞)LvB/K∞L_(v)(B//K_(oo))\mathscr{L}_{v}\left(B / K_{\infty}\right)Lv(B/K∞) be the anticyclotomic BDP ppppp-adic L-function in [3,38].
The ΛQpΛQpLambda_(Q_(p))\Lambda_{\mathbb{Q}_{p}}ΛQp-modules Sv(B/K∞),Xv(B/K∞),(Lv(B/K∞))SvB/K∞,XvB/K∞,LvB/K∞S_(v)(B//K_(oo)),X_(v)(B//K_(oo)),(L_(v)(B//K_(oo)))S_{v}\left(B / K_{\infty}\right), X_{v}\left(B / K_{\infty}\right),\left(\mathscr{L}_{v}\left(B / K_{\infty}\right)\right)Sv(B/K∞),Xv(B/K∞),(Lv(B/K∞)) can be decomposed in terms of Selmer groups and ppppp-adic L-functions of λ,λ∗/χ∗⋅χλ,λ∗/χ∗⋅χlambda,lambda^(**)//chi^(**)*chi\lambda, \lambda^{*} / \chi^{*} \cdot \chiλ,λ∗/χ∗⋅χ. Then, we approach the BDP main conjecture based on Iwasawa main conjecture for imaginary quadratic fields proved by Rubin [43].
The second approach generalizes to CM elliptic curves over totally real field [6,30][6,30][6,30][6,30][6,30].
ACKNOWLEDGMENTS
The author thanks Ashay Burungale, John Coates, Henri Darmon, Yifeng Liu, Christopher Skinner, Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang for their helpful comments, discussions, and constant support.
FUNDING
This work was partially supported by NSFC grant #11688101.
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YE TIAN
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, ytian@math.ac.cn
ARITHMETIC AND GEOMETRIC LANGLANDS PROGRAM
XINWEN ZHU
ABSTRACT
We explain how the geometric Langlands program inspires some recent new prospectives of classical arithmetic Langlands program and leads to the solutions of some problems in arithmetic geometry.
The classical Langlands program, originated by Langlands in 1960s [41], systematically studies reciprocity laws in the framework of representation theory. Very roughly speaking, it predicts the following deep relations between number theory and representation theory:
A special case of this correspondence, known as the Shimura-Tanniyama-Weil conjecture, implies Fermat's last theorem (see [62]).
The geometric Langlands program [42], initiated by Drinfeld and Laumon, arose as a generalization of Drinfeld's approach [20] to the global Langlands correspondence for GL2GL2GL_(2)\mathrm{GL}_{2}GL2 over function fields. In the geometric theory, the fundamental object to study shifts from the space of automorphic forms of a reductive group GGGGG to the category of sheaves on the modul space of GGGGG-bundles on an algebraic curve.
For a long time, developments of the geometric Langlands were inspired by problems and techniques from the classical Langlands, with another important source of inspiration from quantum physics. The basic philosophy is known as categorification/geometrization. In recent years, however, the geometric theory has found fruitful applications to the classical Langlands program and some related arithmetic problems. Traditionally, one applies Grothendieck's sheaf-to-function dictionary to "decategorify" sheaves studied in geometric theory to obtain functions studied in arithmetic theory. This was used in Drinfeld's approach to the Langlands correspondence for GL2GL2GL_(2)\mathrm{GL}_{2}GL2, as mentioned above. Another celebrated example is Ngô's proof of the fundamental lemma [55]. In recent years, there appears another passage from the geometric theory to the arithmetic theory, again via a trace construction, but is of different nature and is closely related to ideas from physics. V. Lafforgue's work on the global Langlands correspondence over function fields [39] essentially (but implicitly) used this idea.
In this survey article, we review (a small fraction of) the developments of the geometric Langlands program, and discuss some recent new prospectives of the classical Langlands inspired by the geometric theory, which in turn lead solutions of some concrete arithmetic problems. The following diagram can be regarded as a road map:
Notations. We use the following notations throughout this article. For a field FFFFF, let ΓF~/FΓF~/FGamma_( tilde(F)//F)\Gamma_{\tilde{F} / F}ΓF~/F be the Galois group of a Galois extension F~/FF~/Ftilde(F)//F\tilde{F} / FF~/F. Write ΓF=ΓF¯/FΓF=ΓF¯/FGamma_(F)=Gamma_( bar(F)//F)\Gamma_{F}=\Gamma_{\bar{F} / F}ΓF=ΓF¯/F, where F¯F¯bar(F)\bar{F}F¯ is a separable closure of FFFFF. Often in the article FFFFF will be either a local or a global field. In this case, let WFWFW_(F)W_{F}WF denote the Weil group of FFFFF. Let cycl denote the cyclotomic character.
For a group AAAAA of multiplicative type over a field FFFFF, let X∙(A)=Hom(AF¯,Gm)X∙(A)=Homâ¡AF¯,GmX^(∙)(A)=Hom(A_( bar(F)),G_(m))\mathbb{X}^{\bullet}(A)=\operatorname{Hom}\left(A_{\bar{F}}, \mathbb{G}_{m}\right)X∙(A)=Homâ¡(AF¯,Gm) denote the group of its characters, and X∙(A)=Hom(Gm,AF¯)X∙(A)=Homâ¡Gm,A